Learn Turkish: Valizi hasar gören yolcu şikâyet ederek tazminat istedi. - The passenger whose suitcase got damaged asked for compensation by complaining.

In Turkish, "whose" is expressed by combining a noun with a possessive suffix and a subject relative clause (the -(y)An form). Here, valizi (his/her suitcase) comes first. Then, the verb phrase hasar gören (which suffered damage) modifies the main noun yolcu (the passenger). Literally, it translates to "the his-suitcase-got-damaged passenger".

Why does the sentence use gören for "got damaged"? Doesn't görmek mean "to see"?

Yes, görmek primarily means "to see", but it is also used in Turkish to mean "to experience" or "to suffer". Hasar görmek is a common collocation that means "to get damaged" or "to suffer damage".

What is the grammar behind şikâyet ederek?

The suffix -(y)erek or -(y)arak is attached to verbs to show the manner in which the main action was performed. It translates to "by doing something" or "while doing something". Attached to şikâyet etmek (to complain), it becomes şikâyet ederek (by complaining).

The English translation says "asked for compensation", but doesn't istedi mean "wanted"?

Yes, istemek means "to want", but it is also the standard verb for "to ask for" or "to request" something. When you want to say someone asked for a refund, compensation (tazminat), or a favor, you use istemek.