Everyone who disagrees is not conservative. Why does this black-or-white thinking always come up? I'm very liberal, and I don't agree with the trans movement either. And it's in fact BECAUSE I'm socially liberal.
See, long ago, society invented this thing called "gender roles". Men should do this, and women should do that. And there are myriad reasons why they were started. At best, it was disciplined rules to help primitive people survive when nature was still extremely harsh and fatal. And at worst, it was used as a way for those in power to keep others from taking their power. But regardless of origins, it was a thing that got passed down into the present day, where there is absolutely no reason for it in modern society. These gender stereotypes still exist only because people saw previous people doing it, so they feel that it's "right".
And then a few decade ago, there was something called a "feminist movement". These are people who felt like these gender roles were causing harm to women. So they advocated to abolish these rules. Women should be free to do anything they want to do, and act however they want to. And society should not call somebody a failure of a woman (or failure of a man) because they do not adhere to these stereotypes. Which sounds fantastic to me. Men and women are free to do whatever they want to, with no barriers in their way.
But now, comes along the trans movement. And suddenly, not only are gender stereotypes important again, but they are somehow THE most important thing in determining if you're truly a woman or a man. Teaching people that if you're a woman but you behave "manly", then, well, you're probably a man. You should just become a man. It's not right to have you acting like this while still being a woman. And I heavily disagree with this logic. I think sex is the only thing that exists, and gender is completely fake.
And yet, like many other people, I don't care if a person is trans. There is no secret desire to "genocide" anyone. Wear what you want, date who you want, act how you want. It's your life. Have fun. I just disagree that the pro-gender ideology is correct and that everybody else must be forced to believe the same.
Like others have just said, there should never be any excuse to cause harm upon those you disagree with. I also don't agree with religious logic, or pro-corporation logic, or people who wear socks with sandals. But at the end of the day, these same folks are also nice people and I can get along with them all just fine without having to try to ruin their lives or make up pejorative labels to stick on them. Thinking that you are "right" is NEVER a justification to attack others.