Except they're given very explicit exemptions, otherwise for example a site specifically aimed at kids couldn't remove pornographic adverts spammed to their message boards without also approving every other post.
Heck if you took that route then by removing posts with content that is illegal in the US, thus complying with other US laws would also make them publishers and so no company, organisation, or individual could host any form of comments online without being a publisher and having all the liabilities required for that.
The result of that would be Twitter would shut, Facebook would shut, even the Republican only sites would have to shut as they'd now be jointly liable for every single post, including the ones that go against their T&C as they'd no longer get the protection.
Your take on this is almost as funny as when we used to get people claiming we were infringing their first amendment rights by removing links to competitors....
The exception you refer to is under 1996 CDA Section 230. Which is what Trump pointed it at his Executive Order today.
Asking the FCC to run investigation if the big 3, Twitter, FB, YT, are complying with the rules of Section 230 within 30 days.
If you bothered to watch the sum up by Tim Pool posted above, explains in detail (having the legislation on the screen) that those 3 cannot censor opinion if it is not legally objective (aka against the law), lewd, harmful, racist etc. Is very specific legislation.
By censoring opinion, they void themselves protection because they do the job a newspaper would do by editing and censoring articles. Hence they become a publisher with the liabilities this implies
That is the problem, that is why the stocks of the big 3 are collapsing tonight because Alex Jones alone can drag them to court now and send them the cleaners. Let alone everyone else, including Trump.
That is why even Pelosi (of all people) came out tonight in support according to Reuters
When asked about President Trump’s expected executive order on social media platforms, Nancy Pelosi did not hold back her criticism of the companies: 'All they want is to not pay taxes’
https://twitter.com/Reuters/Status/1266090615676141568
Zuck runs away hiding because he doesn't want investigation in his company.
https://twitter.com/ReutersWorld/status/1266134659064385537
Hence he came yesterday out in panic mode condemning Twitter, given FB has thousands of examples of censoring and editing posts that are not restricted by Section 230. Hell just yesterday arbitrary deleted a post in a closed forum discussing why the executive orders in the Dem states are so contradictory with themselves. And was banned as "hate speech". Yet there was absolutely nothing that any logic person could perceive as hate speech.