Last April, a high school senior in Utah named Gabriela Merida joined a student town hall hosted by her governor. She introduced herself, noted the pronouns she used and broached the subject of mental health challenges facing young L.G.B.T.Q. people. How, she asked, did the state plan to help constituencies like hers?

“My preferred pronouns are ‘he,’ ‘him’ and ‘his,’ so thank you for sharing yours with me,” Gov. Spencer J. Cox, a Republican, replied. “We want everyone to feel included. We want everyone to feel safe. And we want everyone to understand that they belong.”

One year later, the Republican-controlled Utah State Legislature has passed a measure to bar transgender girls from competing in girls’ sports, overriding Mr. Cox’s veto. His mention of his pronouns has become fodder for right-wing derision and misleading video clips. And deep-red Utah is now at the center of a new fight that is reordering the nation’s politics, roiling its education system — and, for some Americans, shaking their sense of belonging as a midterm election year unfolds.

From state capitals to schools, Americans are increasingly at odds over issues of identity and language, who can play on which youth sports teams and what can and cannot be said in classrooms. These issues are pitting governors against their state legislatures, business leaders against conservative activists and, in some places, Republicans against one another, while Democrats calibrate their responses and some transgender people feel increasingly isolated.

To Democrats and some Republicans, the legislative pushes on these issues amount to an effort to inflame the G.O.P. base at all costs — even when it means children and their families see their governments singling them out.

Much of the policy dispute in the first months of 2022 has centered on two issues: efforts to restrict transgender youths’ health care and participation in girls’ sports, and a sweeping Florida law signed by Gov. Ron DeSantis, a possible presidential candidate. That legislation, which prohibits classroom instruction about sexual orientation and gender identity in some elementary school grades, is called the “Parental Rights in Education” measure — or, to its critics, the “Don’t Say Gay” law.

Such efforts come as parents have spent two years navigating exceptionally difficult questions about schooling, from mask mandates to divisive debates over curriculums. Some parents have also grown anxious about what young children are being taught about sexual orientation and gender identity. At the same time, debates regarding transgender athletes have grabbed headlines and raised questions about how transgender children participate in sports.

At the University of Pennsylvania, a transgender woman named Lia Thomas dominated swim meets and won a national title, drawing criticism from rivals and some teammates as well as from prominent female athletes.

There is relatively sparse nonpartisan polling on these issues, but a survey from the Public Religion Research Institute last year found that while 82 percent of Americans supported laws that protect L.G.B.T.Q. people from discrimination, they were more uneasy about other questions.

Only 36 percent of Americans surveyed said transgender girls should participate with other girls in high school sports. A Gallup poll last May found that 62 percent of Americans said transgender athletes should be allowed to play only on sports teams that matched their gender assigned at birth, though an earlier Marist Poll survey found far more opposition to a bill barring transgender student athletes from sports teams that reflected their gender identity.

In all that turmoil, Republicans see a political opening.

Lawmakers in states beyond Florida have recently signaled intentions to emulate the state’s new law. Opponents warn that parts of the law may have a chilling effect on teachers and on students of all ages, including some who have relied on schools as a safe place to talk about personal issues.

The debate has turned ugly: Some proponents of the Florida law call its critics “groomers” — a term associated with pernicious decades-old smears suggesting that L.G.B.T.Q. people pose a threat to children.

Several states have also passed restrictions on transition care for minors, and the governor of Texas directed state officials to view medically accepted treatments for transgender youths like puberty blockers and hormones as abuse, a policy that quickly became the subject of litigation. On Friday, Alabama’s governor signed legislation that stops medical professionals from providing care that helps transgender young people in transitioning, among other sweeping restrictions.

Supporters of the bills frame the push on girls’ sports — a subject that has appeared in some Republican campaign ads — as a matter of fairness for female sports competitions and part of a broader set of parental concerns, even as Democrats and some Republicans question what real-world problems these measures are intended to solve.

“I hope the left doesn’t understand how big of a deal it is,” said Tony Perkins, the head of the Family Research Council. “They’ll never know what hit them.”

And the Republican National Committee is eager to discuss the schools issue, pledging in a statement that the party would battle “attempts to force conversations around sex and transgender issues on our youngest children,” while also alluding to the committee’s gay outreach initiative.

For a while, the power of the right on L.G.B.T.Q. issues seemed diminished as growing numbers of Americans, including Republicans, accepted same-sex marriage. After an outcry, North Carolina repealed a law targeting transgender people’s use of public bathrooms. And other efforts to limit transgender rights, from Texas to Kentucky, sputtered.

But activists on both sides see this moment as different.

During the pandemic, new sports-related bans have not always attracted the kind of national blowback that the North Carolina bill did. Former President Donald J. Trump rolled back protections for transgender people with his base’s support. And with a solidly right-leaning Supreme Court at their backs, many Republicans are going on offense on cultural issues like L.G.B.T.Q. rights.

The number of transgender-rights-related measures in statehouses has risen significantly.

The Human Rights Campaign, an L.G.B.T.Q. rights organization, said that in 2020, state legislatures introduced a then-record 79 bills that the group considered to be anti-transgender. In the first months of 2022, that number is already at 140, Cathryn Oakley, the organization’s state legislative director and senior counsel, said in an interview last week.

“They just kept pivoting until they could find the thing that they thought would capture the public’s imagination and turn them against L.G.B.T.Q. equality and acceptance,” she said. “They’ve cared about women’s sports for exactly as long as it was politically expedient.”

A year after she spoke with Mr. Cox during that town hall, Ms. Merida, now 19 and a student at the University of Utah, has watched the developments in her state with alarm. As a bisexual teenager who has confronted mental health challenges, she said she had engaged Mr. Cox because she was worried about higher suicide risk among L.G.B.T.Q. youths.

“It’s tragic. It’s devastating, especially for trans youth,” she said. “During these times, you feel kind of helpless. But you keep on fighting.”

President Biden made overtures to transgender Americans in his State of the Union address, and the administration has taken other steps. But some want other Democrats to push back harder.

“Republicans are trying to weaponize fear and ignorance of L.G.B.T.Q. people, and specifically trans children, in order to gain an advantage in the midterms,” said Charlotte Clymer, a writer and transgender activist, who lamented “a vacuum” of information and advocacy from Democrats. “Republicans are all too happy to fill that void.”

Some Republicans recoil from the legislative efforts. In his veto letter, Mr. Cox said that in Utah, there were four transgender children out of 75,000 high school athletes. Just one of them, he said, was playing girls’ sports. (Mr. Cox declined to comment on the veto for this article. Over the weekend, though, he responded on Twitter to right-wing critics, writing, “If you have to doctor a video to make a kind gesture to a nervous kid look bad, that says more about you than me.”) Gov. Eric J. Holcomb of Indiana, another Republican, vetoed a similar bill, though the Legislature may override him.

“I worry about the message we’re sending to trans kids,” said State Representative Mike Winder, a retiring Utah Republican who backed the veto. “And a little bit, we’re a solution looking for a problem.”

In Florida, State Senator Jeff Brandes of St. Petersburg thought he had found a compromise: He tried to amend the education-related legislation to effectively ban sex education through third grade without singling out L.G.B.T.Q. people. When Republicans voted down the amendment, Mr. Brandes said, he became convinced that the legislation was meant to “impact” the L.G.B.T.Q. community and create a political wedge.

Mr. Brandes, one of two Republican senators to vote against the bill, acknowledged that the issue of what sensitive subjects are taught polls well, even if the legislation did not address an actual crisis.

“It made me feel like the purpose of this bill was to get into the news and not solve a problem,” he said. “In 12 years as a legislator, I never had a constituent with a child in K-3 raise this issue.”

Emotions have run especially high in Tallahassee.

January Littlejohn and her husband sued the Leon County public schools after contending that they had been excluded from conversations and decisions involving their 13-year-old, who had expressed feelings that they might be nonbinary.

Ms. Littlejohn appeared with Mr. DeSantis at the bill signing last month.

“Parents are being systematically cut out of critical conversations occurring with their child in school,” she said in an interview. “It’s undermining parents and their authority and basically sending a message to children that parents are the enemy.”

But Rocky Hanna, the county schools superintendent, said the new Florida law had sent a chill through teachers.

“They’re scared to death,” he said. “‘What can we say now? What can’t we say?’ The governor has made clear he’s encouraging parents to sue school districts.”

Teachers “don’t ask for these conversations,” Mr. Hanna continued. “But because of the special relationship that a child has with their teachers, sometimes students confide in teachers.”

In Florida and across the country, there has been a significant backlash to the range of new laws.

“We need to be helping our children catch up with math and reading instead of making our schools political and culture war zones,” said Gov. Roy Cooper of North Carolina, the chair of the Democratic Governors Association, who was propelled to office in part by the bathroom bill opposition. “Democratic governors are going to talk about it that way, I believe.”

Ms. Merida recently reflected on a smaller gesture, after learning that her exchange with Mr. Cox had become a right-wing talking point.

“He only shared his pronouns after I shared mine, to support me,” she said. “It was nice to be affirmed like that.”