Learn Turkish: Kaç bardak su var? - How many glasses of water are there?

In Turkish, a very strict rule is that any noun following a number or a quantity word like kaç (how many) must stay in its singular form. So, it is always kaç bardak (how many glass), never kaç bardaklar.

Since this lesson covers the counting word tane, should I say Kaç tane bardak su var?

No, it is better to leave tane out here. We use tane to count individual items (like "pieces"), but bardak is already acting as the unit of measurement (glasses of water). Using both sounds unnatural, like saying "how many pieces of glasses of water."

Where is the word for "of" in "glasses of water"?

Turkish doesn't use a connecting word like "of" for containers and measurements. You simply put the container right next to the substance. Bardak su literally means "glass water," but translates perfectly to "glass of water."

Why does the sentence use var (there is) if the English asks "are there"?

Because the noun phrase kaç bardak is kept grammatically singular in Turkish, the existence word also stays in its base, singular form (var). You don't need a plural verb when the noun itself is singular.