Learn Croatian: Ovdje nema mnogo žena. - There are not many women here.

Yes, it is! However, the genitive plural ending for almost all regular nouns in Croatian is -a. Because of this rule, the genitive plural form of žena happens to be spelled exactly the same as the basic singular form (žena). In spoken Croatian, the final "a" is stretched out longer in the plural to help tell them apart, but in writing, you just have to look at the context clues (like the word mnogo).

Why do we need to use the genitive case for žena in this sentence?

There are actually two words triggering the genitive case here! First, quantity words like mnogo (many / a lot of) always require the noun that follows them to be in the genitive case. Second, the word nema (there is not / there are not) also forces the noun into the genitive.

The English translation says "There are not", but doesn't nema come from a singular verb?

Yes, nema literally comes from the "he/she/it" form of the verb nemati (to not have). But when it is used to point out that something is missing, it acts as a fixed, "impersonal" word. It means both "there is not" and "there are not". It never changes to a plural form, even when you are talking about multiple things like "many women".

How would I change this sentence to say "There are many women here"?