Republican-controlled state legislatures are proceeding apace with efforts to restrict transgender children from getting transition-related medical care and participating in girls’ sports. But the effort has hit a few snags in recent weeks.

It culminated in Arkansas Gov. Asa Hutchinson (R) last week vetoing the most far-reaching ban to date on transition-related care. Hutchinson did so while arguing that his fellow conservatives should be careful about getting government involved in such things. Arkansas Republican lawmakers disregarded his argument and overrode the veto.

But a new poll suggests Hutchinson might have been on to something — and that Republicans pushing these bills might indeed want to proceed with more caution than they have been.

The PBS NewsHour-NPR-Marist College poll is the first broad look at how the public feels about the bills that have been introduced in dozens of statehouses and moved forward in states including Arkansas, Mississippi, South Dakota, Tennessee and West Virginia. And the big takeaway is this: Republicans broadly oppose transgender children participating in the sports of their gender identities, but they also oppose the government banning it. And they also oppose the government banning transition-related medical care to children.

The poll shows around two-thirds of Americans overall oppose such bills. Opposition is 67 percent when it comes to the sports-related bills like one passed in South Dakota that was later vetoed in part by South Dakota Gov. Kristi L. Noem (R). The number is 66 percent when it comes to banning transition-related medical care to children and 65 percent when it comes to criminalizing providing such care — both features of the Arkansas bill. In each of the three cases, just 28 percent support such legislation.

Interestingly, the numbers among Republicans are virtually the same as among the broader population on each one. The only exception is when it comes to criminalizing providing transition-related care to children, in which case the GOP is somewhat more evenly split — with 38 percent in favor of that and 55 percent opposed.

The numbers are in some ways a little inexplicable, suggesting we’ll have to wait and see what other data show.

For one, the poll shows only between 18 and 20 percent of Democrats say transgender athletes shouldn’t be allowed to participate in the sports of their gender identity (with slight variances depending on the level of competition), but it also shows a higher number — 25 percent — somehow support bills banning it. The lack of a partisan gap on virtually all of these types of bills is also difficult to reconcile, given where the parties are more generally on transgender issues.

Likewise, the gap between the number of Republicans who don’t think transgender athletes should participate in the sports of their gender identity (between 71 and 81 percent) and those who support bills to ban it (only 29 percent) is huge.

Perhaps that’s because they don’t think this is the place of the government. Perhaps it’s because these complex questions are difficult to poll and people might not understand the question.

But it’s hardly an affirmation of government getting involved in these issues. Repeatedly when asked whether they “support or oppose a bill that prohibits” something related to the rights of transgender children, Americans broke more than 2 to 1 in the “oppose” direction.

It’s also possible these issues will take some time for people to arrive at firm opinions on. As The Washington Post’s Will Hobson wrote in a must-read piece Thursday, the issue of transgender athletes participating in girls’ sports is a thorny one even for many on the left. A central issue is whether and how much to require that testosterone be suppressed — especially given such care might not be practical for or available to some, for a variety of reasons including family ones.

The GOP bills, though, largely skip over that debate entirely, moving to ban transgender athletes from participating in the sports of their gender identity under any circumstances. Other bills, like the one in Arkansas, aim to prohibit virtually any ability of children to transition at all. Hutchinson said in vetoing the bill that he supported more narrowly prohibiting sex reassignment surgery, but that going further than that and taking virtually all transition-related decisions out of the hands of parents, children and their doctors was not a conservative position.

“While in some instances the state must act to protect life, the state should not presume to jump into the middle of every medical, human and ethical issue,” Hutchinson said. “This would be — and is — a vast government overreach.”

It’s not entirely clear how truly ascendant that view is in the broader conservative movement. But this poll doesn’t suggest these bills are something the public or even conservatives are particularly clamoring for. And beyond that, Republicans have lots of recent history to look at when it comes to how the public shifted to the left on gay rights at a rather rapid pace. Given that, their starting point in pushing these transgender restrictions doesn’t appear particularly strong.

What is strong, though? Support for the Equality Act, a Democratic-led bill that aims to include gender identity protections in federal anti-discrimination laws. The poll shows Americans supported the idea 63 percent to 32 percent. All but three House Republicans voted against it, with many citing concerns about religious objections.