What does it mean to "twist/pervert" a religion?

It’s in there zero times, actually.

To answer your rhetorical question, and the OP’s explicit one: when someone says that a religion is being twisted or perverted, what it means is that the speaker doesn’t have the balls to say, hey, maybe this whole religion thing isn’t worth the hassle and let’s just all start acting like grownups now.

I know we’re all supposed to think that all religions are all about peace and everything would be great if we could just see the similarities instead of the differences, but it’s not really true. Christianity is about peace and love, sure, but it still ranges from guilt-inducing to smugly condescending peace and love. Judaism and Hinduism are really more about culture. Sikhism is pretty peaceful… for a religion that packs a dagger. Yes, Islam is literally about peace and submission (to be fair, Arabic has like 6 phonemes and they all mean nice things), but it’s a peace won in battle and love between believers. Muslims do believe that Christians and Jews are people of the book and all, but in my experience, that’s just the framework with which to rigorously point out all the ways that those people are screwing it up. Oh, and Baha’ism is just Islam with the death-to-infidels turned way down and the singing turned up, so okay, it’s peaceful.

That aside, if you want to give Islam a charitable reading, you could argue that when one has a new take on an old religion to spread, territory to defend, and five prayers a day to do, one might get a little heavy-handed when rallying the troops. I mean, if Red Sox fans had their own bible, what do you think it would say they should do to Yankees fans? And in reality, the most violent Muslims are the poor young men who have never seen a breast; there’s four reasons to want to kill infidels, right there. And in places like Iran, the guys in charge are all just trying to out-right-wing each other to stay in power – sound familiar?