This is not true. Liberals tend to be calling out powerful people who harbor hateful ideas. It seems like it can be a bit heavy handed at times, but it's generally awareness and boycott actions.

Meanwhile the right was and continues to be book burners, just like their Nazi counterparts. Remember the censorship debates of 1980-2000 (although this was partly bipartisan)? It continues to this day.

It's the right that I've seen consistently try to corrupt institutions and restrict free speech using the government. They're the ones developing lists of "undesirables". The left just says that it's okay to be gay, and gives out free condoms.

There's a little bit from all sides now. Also, harboring some progressive ideals, doesn't make you free from bigotry in all regards. That said, the right pretending that free speech is now a right wing virtue is absurd. It's just as ridiculous to pretend it's "even".