At least that's what I've seen people say, and it sounds quite plausible.
So in that regard there is some way that the web gets policed in the US.
There is an ideological dispute about how big a difference that is.
Many years ago, I identified as an anarchist (not anymore). While I was an anarchist, I followed intently the debates between anarcho-socialists (left anarchists) and anarcho-capitalists (right anarchists). A major part of the debate was whether it made a difference whether someone doing something was the government or a private entity. From the anarcho-capitalist viewpoint, the public-private distinction has a fundamental philosophical, ideological, political and moral significance – governments and private corporations are radically different kinds of entities, whose actions ought to be judged by very different standards. From the anarcho-socialist viewpoint, the public-private distinction is largely artificial, arbitrary, irrelevant – governments and private corporations are just two slightly different flavours of the same soup.
Although few people involved in these debates over the possibility of "censorship by private corporations" would identify as anarchists (of either variety), I think the same distinction is at play here; but, unlike those anarchist debates, the parties in the "censorship by private corporations" debate seem less aware of the role that fundamental difference may have in motivating their respective positions.
By "society", do you mean a loud minority of blue checkmarks on Twitter?