You keep saying this, backed up by no more than your own circle of acquaintances. Here's what happens when proper weighted opinion polls are done. This first table is the results from a poll conducted in 2020.
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A spread of opinions, but the Yes to using wrong-sex toilets becomes a No when it's specified that these people have not had their naughty bits lopped off. (How anyone is supposed to tell whether they have had this done or not was not specified.)
Now here is the same table, but results from a poll carried out at the end of 2024, just four and a half years later.
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The only thing even a plurality of respondents are inclined to allow trans-identified people to do is to "identify" as the opposite sex, but that doesn't get them any actual concrete concessions, as everything else has turned to a big fat No, most of it by a majority not just a plurality. The responses from women (to the left of the overall responses) don't differ much from the overall responses. No to pretty much everything, just a slightly more emphatic No as regards trans-identifying women when the stipulation that they have not had surgery to create a fake penis is added.
What changed in the space of less than five years? The fact that a lot of sunlight has been shone on the reality of trans demands and the entire trans agenda. That the myth of the poor vulnerable flower who has had all the surgery and does the absolute best he can to look like a normal woman and just wants to slide quietly into the Ladies', use the facilities and slide back out again without disturbing anyone has shrivelled and died in the light. The bearded men insisting on their rights, the wanking selfies in the Ladies', the pee protest in Parliament Square where trans people urinated on statues and everywhere else available to advance the cause of their being allowed to use the wrong-sex toilets, all these things leave an impression.
And it is absolutely not the case that women actively favour "sorting by gender identity". American women may tend to vote Democrat, but people have lots of reasons for choosing how to cast their vote and claiming that a vote for that party is a positive endorsement of a move to allow men into women's intimate spaces is disingenuous in the extreme. Ask American women questions like the above and see where you get. Then come back in 2030 and see if they're still in the same place.