This is basically entirely a result of the new prominence of the transgender movement.
A lot of people came around to acceptance of gays because they were convinced by arguments that "gays just want the same rights as you, there's no slippery slope"
but now we see that it's a mainstream liberal position that children should be given sex change hormones and puberty blockers.
There obviously was a slippery slope and people feel like they were tricked into supporting gays under false pretenses.
progressive cultural shifts engender backlash. advocating for social equality and acceptance for ethnic, religious, and sexual minorities wakes people up. seeing opposition to these movements should inspire a greater push to advance equal rights and acceptance, not cowardice and retreat (like mortimer's explanation).
if someone was supportive of homosexuality, but then heard about transgender people and got turned off, then lemme say this: i understand the hesitance to accept and understand people who are different from you. especially when those ppl are different in a way that seems new, that you're unfamiliar with. but the challenge for all of us, morally, is to have open minds and seek to understand people who are different from us.
the lgbt movement pushed for acceptance of gays and lesbians. they were on the right side of history and morality in the 50s and 60s, and they still are. transgender people have only pushed for acceptance more recently. understandably, more people are hesitant to accept them. but with greater understanding, things will change. and ppl pushing for transgender acceptance are on the right side of history and morality too.
there are differing opinions within the lgbt movement and within the left about how to understand and accept transgender people. most want to be open minded and sensitive to the experiences of those who feel differently with their gender identity. children understand their gender identity at the same time, regardless of whether they're transgender or not. children who are transgender should be treated appropriately with the consultation of parents and doctors. treatment to help them live their true gender identity may be necessary, especially to treat gender dysphoria (distress/harm caused by gender-sex mismatch).
regardless, for every theoretical parent who gives their child hormone therapy and puberty blockers, there are dozens of transgender children and young adults who kill themselves because their families reject them or kick them out. it's imperative that regardless of how you feel about treating these kids, they are real; being transgender is real, natural, and kids should be accepted and loved.