After reflecting on this, I'll take a moderate stance and say both sides are stupid. Which is to say, there are people who are extremely sensitive (they have no social hardiness and become emotionally wounded at the drop of a hat), and there are people who are extremely harsh in their language (for example, people who intend to belittle and break down a person). These extremes are both shitty and both contribute to the problem.
I'll take the "victim blaming" angle and use the example of a mugging. You're out at night, in a bad neighborhood, alone. You pick a dark alley to walk down with your designer coat and bewildered expression. Some guy peels himself off the shadows, says he has a gun, says to drop your coat and wallet and scram. While he's the one committing the crime, you could have done something to mitigate your risk. If we were talking percentages I'd say the victim is at least 10% at fault here, possibly as much as 25% depending on just how many muggings happen in that neighborhood.
Go ahead and use the example of the girl who goes to a bar she doesn't know, alone, gets super fucking drunk, and some awful criminal takes advantage of her vulnerability. You know what? Maybe she shouldn't have gotten so drunk she fucking blacked out. This isn't "she dressed provocatively so she's totally at fault." This is "I drove up into the snowy mountains with no tire chains, slid off the road in the snow, and died. If I used chains I would be alive today."
So let's look at someone who has a gender identity that does not match their physical gender. I see it bandied about that gender-normative people have an unearned advantage in society; I'd return that transgender people have an earned disadvantage. Who you are inside, your real identity, is very different from your public, perceived identity. If you decide to display a public identity that is far from the expected norm in a culture you should expect that there will be some cultural backlash. If I split my tongue and get a full-body-and-face tattoo of scales and say I'm a lizard man, perhaps I will experience some difficulty finding employment in many fields. That is the choice I make when I display a non-standard identity. I earn disadvantages.
I'd agree that these cultural backlashes should be more moderate. There's no excuse for violence, or for denying someone housing or employment, because of their appearance, self-identity, etc. People need to live. But if a player shows up for my D&D game and speaks only a screeching pig latin while grinding his butt on the table (because that is how he identifies himself), you may excuse my narrowminded wish to never see him again and refuse to invite him back to the game. I just uninvited two players because they were fucking weird (and in one case brought a large folding knife to the game despite me specifically mentioning my house rules of "BYO snacks, don't be drunk or high, no weapons"). People make compromises about who they are every day in order to fit in better, because they value the company of others more than exposing that last bit of personal identity. If you really value your personal identity, you're free to trade the fullest expression of that for only having acquaintances who are just like you.
Now let's talk about comedians. You'll see some white guy get on stage and talk shit about X group of people. I think that's wonderful. He has the right to free speech, and if the owner of the venue doesn't want him there he can ask him to leave, and if the patrons hate the speech they can boycott the club until the owner changes his policies. And if there aren't enough patrons who have a problem with it, then the sensitive lillies can go to a club where they only serve positive uplifting comedy.
People will draw a connection to comedians making Jew jokes in Nazi Germany, and how that might have affected the population's views toward Jews. Let me tell you, people are gonna be racist or not regardless of what a comedian tells them. Please try to argue that a few comedians in a few clubs were able to incite the German people to support a Holocaust. Comedians are not that powerful; they frequently just tell you what you were thinking anyway but you can't get away with saying. The person on stage saying something socially unacceptable creates tension which we release by laughing, by bonding with the other audience members as a temporary tribe of people who agree that this guy is fucking hilarious and that's so true and oh man did he say that? You walk out of the club and you are no more racist or sexist or whatever than before.
I will agree that continued exposure to ideas will change how you think and how you feel. But you need to seek out those experiences or they need to be pervasive in your culture for them to have a strong effect on you. This is why intolerance of minority races and ethnicities, and of non-normative gender and sexual roles, and of minority religions, is maintained. If you grow up and your parents are racist, and your school friends are racist, and your politicians and shamans and secular teachers are racist, you have a greater chance of growing up racist. Nurture can be a strong MF force in your life.
Hence the plaintive cry of the downtrodden to "stop hate speech" which really amounts to "everyone everywhere in this culture needs to stop saying anything I don't like", which is kinda the only way we're going to change some of these very negative cultural backlash behaviors. You need to harshly stigmatize negative speech. But it's a minority group trying to impose what amounts to censorship on the majority group. Few people exist in the Venn diagram space of being only a minority, which means almost all people need to be censored at least a little. And this ignores the diehard intolerants out there who will actively maintain their intolerant speech. And you also have many people who simply hate censorship and say "fuck off and mind your own business" when they're commanded by a powerless minority.
Cultural change is very difficult.