Turkish is the adjective that belongs to the country Turkey. It covers three things at once: where something comes from (Turkish coffee), who a person is (a Turkish student), and the language (a Turkish newspaper). It always starts with a capital T, wherever it sits in the sentence. For a person English gives you two ways to say it: a Turkish woman uses this adjective, and a Turk uses the noun. It does not take more or most, because a thing is either from Turkey or it is not.

Here Turkish is a noun: the name of the language. Language names in English take a capital letter and no article, so you say I am learning Turkish, not the Turkish. They have no plural either. To say what language something is written or spoken in, English uses in: in Turkish. Notice that the same word is doing two jobs across the page — as an adjective it can point at the country, but as a noun it only ever means the language.