And there you have self censorship at work!
4chan is an anonymous, interest-based forum that is more or less irrelevant to most non-internet people.
~20% of Americans use twitter. (https://www.pewresearch.org/fact-tank/2019/04/10/share-of-u-...)
There's definitely some influence there, but the twitter's importance and impact on the world is severely overhyped.
I'm not sure I would make the same claim. Twitter is wildly popular among journalists, pundits, and other members of the media. Due to its outsized importance within these groups of people, I can easily see Twitter playing a large role in shaping media narratives.