“Are you watching Turkish series?” sounds ungrammatical to native English speakers. Series is countable. However, there’s really no way to mark the plural of series (“serieses” is not a word.) So we’re left with this word with no determiner, apparently singular, and that makes it... just wrong. In the same way the sentence “Is there book on the table?” is wrong.
“Are you watching the Turkish series?” would be correct here (which version of a franchise that has had both Turkish and English-language productions?)
JSNuttall is correct. There's nothing wrong with the English sentence. "Series" is one of those words that doesn't have separate singular and plural forms. As native English speakers, we know instinctively that "series" is plural in this sentence from the context, because there is no article in front of the word. There are a lot of words like this in English that don't have separate singular and plural forms, like "species," "sheep," "aircraft," "fish," "deer," "moose," etc.
I disagree; if "series" is taken as plural, you absolutely don't need the article; in fact, the sentence sounds more natural without it. Do you think "Are you watching cartoons?" is ungrammatical? If not, why would you need the article for "Turkish series" but not for "cartoons"?
There are 2 words: series and serial and they are not the same. A series contains the same characters throughout, but each episode is a different story. So you can start watch series from any episode. But if you want to watch a serial, you should start from the first episode because each episode is a continuation of previous one. So I think that "Turkish serial" is a more appropriate trnslation.