Is censorships a legitimate word? Obviously it could be used to mean multiple censorships for something.
The OED has a citation from Thomas Arnold writing in 1840 in which he refers to the censorships of the two Scipios. That is, admittedly, a rather specialized use, but in 1909 Shaw wrote
Persecutions . . . are trifles compared to the mischief done by censorships in delaying the general march of enlightenment
while a twentieth century citation has
The repressions and censorships of which Freud and others make so much are connected with changes in the nervous system.
So, regardless of what you may mean by ‘a legitimate word’, there is good evidence of its use, even if it may not be frequently found.
Of course this is a legitimate word.
Some more quotes below.
In Mrs. Warren's Profession, Shaw wrote that "all censorships exist to prevent anyone from challenging current conceptions and existing institutions" (The Author's Apology).
Dr. Sue Curry Jansen, an expert on issues of freedom of expression, talks about constituent and regulative censorships in her monograph, Censorship: The Knot that Binds Power and Knowledge (OUP, 1991).
Remember, you can use virtually every English noun as countable or uncountable.
That is why David Crystal wrote in The story of English in 100 words, in chapter 34 "Information(s)",
The message is plain. Words can be countable or uncountable depending on the sense we have in mind when we use them (Crystal 2011: 90).