Conservatives are not against gay or trans people or just anything new. Conservatives actually stand for conservation of democracy institutions: free speech, private property, rule of law and, of course, personal freedom with "being gay" being of the many examples.
Now i dare you to name one government that is conservative as per definition above. It is definitely not in EU, because they all are quite left. "евросовок" is a word for last twenty years for a reason. I have no idea but i would bet you that there is zero gatekeeping for HRT meds in profoundly conservative texas (for adults).
You could argue that conservative people stand against LGBT-whoknowswhat-movement and you'd be absolutely right. That is because LGBT-movement was hijacked by leftist as a very effective vehicle to victimize people (and literal children) as per cultural marxism doctrine. Surely it would take little effort from you to find some gay/trans people that are against LGBT-movement. I'd recommend to hear their side out if you really want to talk politics, not blab about.

Unfortunately this whole nuanced discourse collapsed (despite being old and well-established) with all core terms being redefined-oversimplified under leftist propaganda.
picrelated https://i0.kym-cdn.com/photos/images/facebook/001/360/162/f4e.png