Can you point me to some literature on that? The infamous law in England that all plays had to be approved by the Lord Chamberlain (a situation which persisted until the late 1960s) dates from 1737, and thus over a century after Shakespeare’s time.
Generally, though, the monarchy of the time was quite supportive of the theatre. Just as well really, as the City of London was run by Puritans at the time and they'd gladly have shut down every one of the theatres. That's the reason they were generally built outside the City.
That moral defender said that they don't employ censorship. Censorship is something the government does. No, they were going to be censored by private entities... how the turn tides... it feels like going back in time and not only because Americans began to talk about culture.