The last time Republicans in South Dakota made a serious push to bar transgender girls from school sports, in 2019, their bill was known only by its nondescript numerical title, Senate Bill 49. Its two main sponsors were men. And it died without ever getting out of committee, just 10 days after it was introduced.

But when Republicans decided to try again in January, they were far more strategic in their approach. The sponsors this time were two women who modeled their bill after a template provided by a conservative legal organization. They gave the bill a name that suggested noble intent: the “act to promote continued fairness in women’s sports.” Supporters from Minnesota and Idaho traveled to the Capitol in Pierre to testify that a new law was urgently needed to keep anyone with male biological characteristics out of female competitions, even though they acknowledged only a handful of examples of that happening in South Dakota.

“These efforts appear to be far more slick, and far more organized,” said Elizabeth A. Skarin of the American Civil Liberties Union of South Dakota, which opposes the bill. “Anytime they give a bill a name in South Dakota,” she added, “you know something’s up.”

Then things took an unexpected turn. Gov. Kristi Noem, who is seen as a possible contender for the Republican presidential nomination in 2024, demanded changes to the bill before she would sign it. The response was swift and harsh: Social conservative activists and Republican lawmakers accused Ms. Noem of being cowed by pressure from business and athletics organizations, which have been successful at stopping laws in other states that single out transgender people for exclusion and feed ugly stereotypes.

In Mississippi and Arkansas, they are set to become law this summer. And similar bills have been introduced by Republicans in two dozen other states, including North Carolina, where an unpopular “bathroom bill” enacted in 2016 prompted costly boycotts and led conservatives nationwide to pull back on efforts to restrict rights for transgender people.

“You make change in our society by making laws, and luckily we have some great states that have stepped up,” said Beth Stelzer, the founder of a new organization, Save Women’s Sports, that she said opposes “demolishing women’s sports for the sake of feelings.” Ms. Stelzer, an amateur power lifter who was in North Carolina this week for the bill introduction, has also testified in support of new laws in South Dakota, Montana and Arkansas.

“women’s sports as we know it will die’’ if transgender athletes were allowed to compete.

requires athletes who are transitioning to female to be on testosterone suppression treatment for a year before they can compete on a women’s team.

Ms. Stelzer, who competes in a weight lifting league that does not allow transgender women to participate, says the point is to get ahead of what she and other activists believe will become a bigger issue. “We’re nipping it in the bud,” she said.

In high school sports, where conservative activists have focused much of their attention, policies vary widely. Some states pose no barriers to transgender athletes; some have policies similar to those of the N.C.A.A. that set guidelines around hormone treatment; others have outright bans or demand that students verify their sex if questioned.

Rarely has an issue that so few people encounter — and one that public opinion analysts have only recently begun to study in depth — become a political and cultural flash point so quickly. The lack of awareness creates an environment in which the real impact of transgender participation in sports can be overshadowed by hyperbole.

But the debate also raises questions — that ethicists, lawmakers and the courts are only beginning to address — about whether the decades-long effort to give women and girls equal opportunities in sports is compatible with efforts to give transgender people equal opportunities in life. A lawsuit in federal court in Connecticut brought by three high school runners who lost in competition against transgender girls will be among the first to test how nondiscrimination laws apply.

A mix of factors has helped social conservatives breathe new life into the issue: activists who agreed to give up on unpopular bills regulating public bathrooms; an awareness that women, not men, could be more persuasive and sympathetic advocates; a new Democratic administration that quickly moved to broaden and restore rights for transgender people that the Trump administration had eliminated; and a political and media culture on the right that often reduces the nuanced issue of gender identity to a punchline about political correctness.

Activists who have been fighting the anti-transgender efforts in legislatures and through the courts say the focus on school athletics is creating a false and misplaced perception of victimization.