You might take a look at Chapter VII of Bollobas. In particular,
Theorem VII.1.7 -- which is simple enough that he doesn't bother providing a proof -- states that the expected number of -cliques in is, setting and ,
Also, Theorem VII.3.7 states that if then with probability tending to one, contains no -clique, whereas if then with probability tending to one does contain a -clique. I know this doesn't fully answer your question but it may help.
Incidentally, (you probably already realize that) it is a priori possible (though I don't think it is the case) that, for example, for all , since all we really know by Turán is that
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