I guess this is less about this sentence and more about generic grammar, but I'm going to masquerade it as contextually fitting :) :
So in addition to this phrase, if I wanted to ask someone if they spoke English, does Norwegian grammar allow for the sentence "Jeg snakker engelsk, snakker det du?", or is it different, as in French when we would place the direct object pronoun before the verb?(i.e. "Det snakker du?" or "Le parlez-vouz?/Est-ce que vous le parlez?" where le is the dop and parlez is the verb)
You are partially correct! You use "tyrkisk" when saying something like "Jeg snakker tyrkisk" because it's the noun; in that instance, it doesn't have anything to do with the versions of the adjective since it's a noun.
Having said that, you are correct that you use "tyrkiske" with plural nouns. But! there are instances where you will use "tyrkiske" with a singular noun. You add an -e to the end of adjectives when you are talking about something plural, when it is used along with a possessive pronoun (e.g. min store hund) or when it is used with a definitive or demonstrative article (e.g. (den store hunden).
This is something you'll learn more about later in the course!