is this correct?
''I haven't got curly hair.''

See a translation

What you have written isn't necessarily incorrect, but it's not typically said in this manner. Rather, in US English, this would be said as "I don't have curly hair."

This is because got typically denotes things you physically obtained in the near present. (Comparable to the verbs 'grab' or 'obtain')
--> I got the raise.
--> She got herself a pie from the store.
--> I got us a white wine for the picnic later.

On the flip side, have details things you simply 'have' in your possession. You don't have to have obtained them. You could be born with them or have 'gotten' them in the past.
--> I have a dog named John.
--> I used to have a cat too.
--> Do you have a calculator I can borrow?

Hope this is helpful!!