Learn Turkish: Meşgulüm çünkü ödev yapıyorum. - I am busy because I am doing homework.

It works exactly like "because" in English! You start with the main statement (meşgulüm - I am busy), add çünkü, and then give the reason (ödev yapıyorum - I am doing homework). It doesn't change the word order of either part.

Is çünkü an exception to vowel harmony?

Yes! By standard rules, a word wouldn't normally mix ü and u sounds this way. Çünkü is an old borrowed word, so it keeps its fixed shape. You never have to change its vowels to match other words in the sentence.

Let's break down meşgulüm. Why does it end in -üm?

The base word is meşgul (busy). To say "I am busy", you add the "to be" suffix for "I". Meşgul is a slight exception to vowel harmony rules because it is a loanword, so it takes -üm instead of the expected -um.

Why is there no word for "my" or "the" attached to ödev (homework)?

Because the speaker is talking about the general activity of "doing homework", not a specific assignment. In Turkish, general, non-specific objects like this just stay in their plain dictionary form and sit directly in front of the verb.