1) Getting loose, being wild and potentially engaging in sexual activity with members of the opposite gender (or the same gender if thats what you're in to)
2) Acting crazy due to consumption of large amounts of alcohol, marijuana, molly or other drugs
It can be used as a noun where it is spelled "turnt up" instead of "turned up" as grammatical conventions would suggest. Turnt up is used to described the state of being wild and crazy like someone would be at a party.
It is not to be confused with the vegetable, the turnip, to which it shares some phonic similarities but is otherwise entirely unrelated
Dude 1:Mary's party last night was turnt up.
Dude 2: I know. I had like 7 drinks, hooked up with 2 chicks and was going HAM on the dance floor. I was all the way turnt up.
"You're girl spinnin' on my dick like a CD-ROM"
All the Way Tunrt by Roscoe Dash