Series is both a singular and a plural noun. It can translate to both dizi and diziler. So you have some alternatives:
1) Türk dizisi izliyor musun? - Are you watching (a/any) Turkish series? (singular)
2) Türk dizileri izliyor musun? - Are you watching (any/some) Turkish series? (plural)
3) Türk dizilerini izliyor musun? - Are you watching (the) Turkish series? (plural and specific)
The third one is actually grammatically not a correct translation because there is no definite article (the) but it's colloquially correct. I would actually prefer it to be translated as the second or the first one.
Let's look at the suffixes.
1) Türk dizisi : dizisi only have the possesive ending -(s)I because it's a compound noun.
2) Türk dizileri: dizileri first has the plural ending -lAr. and then it has the possessive ending -(s)I
3) Türk dizilerini: dizilerini has respectively the plural ending, the possesive ending and plus the accusative ending -(n)I. The accusative ending usually translates to the definite article (the) so in this sentence it doesn't make sense in terms of grammar but as I said it's a colloquial translation.
“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?)
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.