"The Turkish professor is not coming to class today."
Translation:Türkçe profesörü bugün derse gelmiyor.
It is unclear in English (and there is no way to clarify), but it is crystal clear in Turkish. It is a professor of the Turkish language :)
Edit: I contradicted myself in my explanation. You could say "The Professor of the Turkish Language" or "the Professor of Turkish," although they both would sound a little clunky in the English version of this sentence.
Here is how I understand it:
Türkçe profesörü: has this ending because it is a compound noun=Turkish-language-professor. Turks see the phrase as something more than its parts; you just have to learn which ones work this way as you encounter them. Türk profesör: has no special ending, because here it is simply an adjective+noun = Turkish professor (like "blue house" or "big horse")