I wouldn't worry about "The Government" shutting down web sites, a bit too obvious.
Rather what happens now is "Private Companies" which just happen to be big multinational corporations "Decide" or are "told" by various entities ranging from SJWs to LGBTQ+ warriors, Political party operatives hoping to rig elections/primaries/get rid of opponents, to Three Letter Agencies that certain content needs to vanish because it's "hateful" or "not inclusive" some such thing.
Full on censorship is already well in swing, just your precious "private companies" are the face of it.