It would be more accurate to say that the conservative movement is presently engaged in a concerted effort to associate transgender in the public mind “with peeping toms or rapists or something”.
Up to at most a few years ago, popular attitudes to transgender women were typically dismissive and mocking (“shemales”, “ladyboys”, “trannies”, “chicks with dicks”), etc., but they weren’t perceived as scary or dangerous to women and children.
At worst, heterosexual cisgender men might be worried that they would somehow be “tricked” into sexual activity with a “tranny” who would then deserve to be beaten up or otherwise punished for “his” duplicity. But nothing in these stereotypes suggested that (cis) women or children were under any sort of sexual threat from transwomen, who were commonly thought of as “gay men masquerading as women” in order to ensnare heterosexual men. (Cf. all the arguments about the 1992 film The Crying Game nearly a quarter-century ago.)
Transmen weren’t even on the radar of popular consciousness back then (with an occasional blip such as the 1999 film Boys Don’t Cry, about the 1994 murder of transman Brandon Teena). They certainly weren’t perceived as posing any kind of realistic threat to anybody, because they were “just women pretending to be men”, and consequently the natural victims of sexual violence rather than perpetrators of it. As the investigating sheriff remarked to Brandon Teena after he was assaulted by the men who later murdered him:
And now, desperate for issues to stoke popular outrage, the conservative movement is trying to repurpose these derogatory “weird pathetic freak” stereotypes as objects of fear, with the specific purpose of drawing popular attention to transgender people and fostering the notion that the public needs to be protected from them (by whom? by conservatives, of course).
Never mind that the best they’ve managed to come up with in terms of actual “protective” policies is to insist that transwomen (who, remember, are traditionally stereotyped as “gay men masquerading as women to seduce men”) belong in the men’s restroom, and transmen (beards, big muscles, and all) belong in the women’s room.
This isn’t about “protecting” anybody, this is specifically about frightening people by putting transgender people in situations where their transgender identity will be apparent and will make transphobic people feel uncomfortable and insecure.