I've just checked a few of my essays from last year.
My short essays had between 5 and 12 sources on the bibliography. But the one with 12 got a considerably better mark than the one with 5, if I remember rightly. Still, I tend to get 2:1s, so even an essay with 5 sources clearly can work ok.
I think it really depends on what the essay title is and what books are available. Sometimes, you find books that directly address the question you're dealing with, and so you only need a few. But sometimes you have to trawl through loads of books and articles, to pick out anything relavent to the title.
I can't believe your lecturer wants you to use 20+ sources for a short essay! Surely the essay would end up as nothing but quotations, with no room to develop your ideas? I just checked the bibliographies of my long essays (4500 words) from last year, and one had 13 sources, the other had 18.