Do not think of the verb as only the word "turn" or else it can seem very confusing. It is easier to think of this verb as "to turn over to". It means to "hand over to" or "to give it to" or "to pass it on to". Another use/example: He was a criminal who decided to turn himself over to the police.

I think that "turn" in this situation is "give", but the meaning of this gift is not something to happy about. Like: he has new car then he don't need that old car anymore so he give the old one to someone else. I don't know how to put it but it's similar to when working in a group and I think this task is quite of hard so I "turn" this task to some one and I get the easy task. Correct me if I'm wrong.