This is just demonstrably untrue. For nearly a century the Soviet Union succeeded by doing exactly that. They had international support from the progressive types too.

Progressives I know are pretty tolerant. It's the conservatives that seem obsessed with free-speech-for-me-and-not-for-thee. Xitter is the loudest example.

But at the same time, both the progressives and the conservatives who are active on political social media (take your pick of platform) are very likely to actively attempt to silence the opposition and punish them for speaking.

It's less a political divide and more that most people are still tolerant of dissenting speech, so the people you know in person will tend to be tolerant. There's a loud minority that's vocal on the internet on both sides that advocates for silencing others.

Which side is often going to court (and losing) to dispute facts (like election integrity or sexual assault allegations)?

Donald Trump was re-elected. He has said that we should deport pro-palestinian protestors on college campuses and has sued multiple news outlets, both on tv and in paper, for their coverage during the election season. It's really hard to find any political figure who is more aggressively targeting speech he doesn't like than Trump.

The Republicans in charge of two school districts near me have been trying to organize book burnings for the last two years.

Get back to me when it's the Democrats.

I keep hearing about Republican book bans, but I've only heard they don't want certain books to be available to children in schools, not that they should be banned in general. Compare this with liberals who got some Dr. Seuss and other books cancelled and removed from Amazon etc.

It's seems like both sides attempt to decrease accessibility to literature that they find objectional, but neither has achieved an actual ban.

https://en.wikipedia.org/wiki/Book_burning

I guess if you think this is fine then that's what you think.

It's not about "what I think is fine." It's about equal rights to speech.