Censorship of Conservative Movies occurs in a variety of ways. The most obvious is to deny movie awards to them, despite their popular and critical success.
Here are examples. Please add to this list:
Grease
Grease was a spectacular hit with A-list actors and actresses, containing a mixture of comedy and fabulous music. It quickly became the highest grossing musical movie as of that time. Its soundtrack was the second highest selling in the United States in that year.
But the themes of Grease were conservative, and it was denied any Academy Awards. Shockingly, it received only one Oscar nomination, for best Original Song (John Farrar, Hopelessly Devoted to You).