Real Dictionary

turn up
phrasal verb: If you say that someone or something turns up, you mean that they arrive, often unexpectedly or after you have been waiting a long time.
phrasal verb: If you turn something up or if it turns up, you find, discover, or notice it.
phrasal verb: When you turn up a radio, heater, or other piece of equipment, you increase the amount of sound, heat, or power being produced, by adjusting the controls.

turn up in American English
to fold (material, a hem, cuffs, etc.) up or over in order to alter a garment
to fold or bend back or over upon itself

turn up in British English
to arrive or appear
noun: the turned-up fold at the bottom of some trouser legs

turn-up
The turn-ups on a pair of trousers are the parts which are folded over at the ends of the legs.