What kind of structure is it?

[Truman]

Originally Posted by philo2009

Your confusion is understandable: this is a highly elliptical construction representing 'a doctor who has turned into a pop star'. It is often written with hyphens, (a doctor-turned-pop star).

Omission of articles, however, in appositive phrases is very common.

If it has something to do with ellipsis, I have to study that again. I don't know well about it.

[Truman]

Originally Posted by konungursvia

I won't tell you a third time. Believe what you will. Learning English by memorizing rules doesn't work: you always forget, or do not know, the exceptions. There's really no choice-- you have to use your head, not your textbook.

English is so called English. English is grammatical but Chinese is not.
Actually "doctor turned pop star" is very "Chinese", no articles no consideration for the voice. And it doesn't matter how many verbs in one sentence. I speak Chinese. I just can't do the same with English or I get low marks.

[konungursvia]

I suggest just enjoying using the language, talking to English-speakers. There are loads of them in Shanghai. That will give you a better grasp than consciously fumbling with thousands of rules. ;)