I left the Mormon church because of its stance on homosexuality (I am a gay male). Their stance on this probably been elucidated upon by someone else, but I'll give some context anyway.
The Mormon church claims to support and love homosexual members- and from their twisted perspective, they're not wrong. If you are mormon and gay (though they don't like it when you identify as gay, just "struggling with same-sex attraction"), then as long as you never have sex or any form of intimacy with someone of the same sex, you can do all the stuff regular mormons can, from going to the temple and marrying a girl you aren't attracted to, and eventually reach super VIP mormon heaven where god will "fix" you and you'll be attracted to girls and stuff. (Or be attracted to guys if you are a lesbian, and so on). That last part isn't a consistent part of the Mormon dogma, I can't recall ever hearing about being "fixed" in the afterlife when I was still in the church; I didn't hear about that part of Mormon theology until after I left the church.
TL;DR: the church is fine with gays so long as they don't act on it
Putting aside how utterly fucked up the idea of "fixing" someone's sexual orientation is, as well as the deprivation of the human right to intimacy and love, as well as the cultural hyper-repression of sexuality in Mormonism, insofar as a top Mormon leader saying that there are no gay mormons (whether or not you define yourself by your sexuality is up to you, but nobody can make that fucking call for you)- aside from all of this, I used to be more than willing to bend over backwards for this religion. It took a long time for me to realize that I was in a bad spot in life because of the church, not because I was a sinner who looked at gay porn. So I decided to take a break from church and focus on my studies instead. Thats when I learned the facts. About Joseph Smith's Polyandry and marriages to underage girls, about Brigham Young's overt racism, about the fact that there is literally no evidence supporting anything that happens in the Book of Mormon, about Joseph Smith's history as a con artist, about his 'translation' of the book of Abraham (the community pooled together some money to purchase some Egyptian manuscripts; JS claimed they were written by Abraham, and translated them in a text called the book of Abraham. A couple years ago, scholars recovered the manuscript that JS translated- they were nothing more than funerary texts. Suddenly the Church claimed that JS didn't translate the texts, he "translated" them: using the texts as inspiration to receive revelation from god. Uh-huh.) The list goes on and on and on, from rituals stolen from the freemasons, to a rite where people promised to literally disembowl themselves if they ever talked about what goes on in the temple, from Joseph Smith ordaining black people even though Brigham Young would later refuse them from joining the church, to claims that the skin color of Native Americans and Africans is a result of a curse from God.
Though disassociated from the church, I was astounded to learn the truth.
The church likes to claim that people who leave are apostates who just can't stop from sinning. That wasn't true for me. I really wanted the church to be true- and would have once been celibate for the rest of my life to prove me devotion to that faith. I couldn't stand for such systemic racism, sexism, bigotry, homophobia, and so on. That's why I'm not a Mormon.