Trans rights campaigning frequently get compared to gay rights campaigning but its very different. Gay rights campaigners were fighting for equal rights which they did not have at the time - the right to be in a consenting relationship, marry, adopt etc. This isn't something that out another group at risk or took away someone else's rights. Trans rights activists can't campaign for this because they already have these rights. They want additional rights which do pose a risk to other groups. The right to self identify as a man or woman and then for this gender identity to allow them access to single sex spaces, into women's sports etc. This effectively removes safeguards put in place to protect women, which were hard won. When single sex spaces were made, it wasn't just the dangerous males who were kept out. But in order to keep the dangerous ones out, ALL males must be excluded from those spaces, including now the ones who say they are women. They still have the right, as does anyone, to use the facilities for their sex, but activists are asking for access to both spaces. This is not fair. It's not a case of a petty "you can't be in my club" but a legitimate concern that gets dismissed as "calling all trans women sex offenders" if it gets voiced.
The focus is often on trans women in these debates and that is simply because women who identify as men don't pose the same potential risks to biological men as the other way around, which is why we needed single sex spaces in the first place.
There is also the access to treatments, especially for children and teens - it is "transphobic" to suggest that they should explore their feelings as to why they want to transition with a therapist of some kind before doing potential harm to their bodies, apparently akin to conversion therapy for gay people. This should be a necessary safeguard - and to be honest is particularly galling when adult women are being denied access to sterilisation and in many places abortion on the basis that they can't know what they'll want in the future.
Also the changing of language - people with vaginas, people who menstruate etc. This makes language so much less clear and means that people who don't speak English fluently, those for whom it is an additional language, and those whose literacy level is poor, essentially making their ability to access vital services even more difficult - why should their access be compromised? Medical records should reflect the biological sex too, not their gender, otherwise transgender people will struggle to access appropriate services.
Part of the problem is unless you are pro ALL trans rights/demands there's a very good chance you'll get labelled as transphobic. Which then contributes to anti trans feelings. I won't pretend to understand gender identity (I'm a woman because I am, not because I identify as one, and I cannot get my head around non binary identity) and I feel utmost sympathy for those with dysphoria. I am absolutely pro people being able to live as however they want to present, I will happily use preferred pronouns or names and of course they shouldn't face discrimination. If the campaign was for third spaces id be totally in support. I'm against abolishing single sex spaces, self ID for access to spaces or as the sole basis for treatments, and the unnecessary complication of language.