Translation:I applied to the department of chemistry.
"its department" is "bölümü"; the -n- is a buffer consonant between the possessive ending and the case ending. ("Bölümün" would be genitive case, "department's", not possessive, "its department".)
Another way of saying it in English is "the chemistry department" -- perhaps this makes it clearer that this is not a (fully) possessive construction but more of a noun-noun compound.
I suppose kimyanın bölümü might be "the department of the chemistry".
It makes sense in British English! I don't understand why this course is so insistent on only accepting American English all the time. None of the other Duo courses are like that. It makes it doubly difficult for anyone who isn't American because we have to remember not just the Turkish, which is hard enough, but also the American phrasing. How hard would it be to just approve alternative translations???
It does make sense.
Longman Dictionary specifically includes "make an application" in its collocation box. https://www.ldoceonline.com/dictionary/application
"Make an application" shows up twice in Oxford Dictionaries' example sentences. https://en.oxforddictionaries.com/definition/application