I'm a left leaning libertarian, and this is one of my hot button issues. My understanding is that liberals (and Democrats) were the free speech vanguards. They protested and promoted free speech in places like the U of California.
My understanding is that conservatives were the book banning book burners. You now have passages in classic books being rewritten by, regrettably, people who are erasing history and quality literature. I learned recently that Tom Sawyer and Huck Finn will be rewritten. It's really astounding.