Turnt (Turnt up) is a slang word which means to be high, drunk, excited, etc... Someone who's drunk or high on drugs might say "I'm Turnt" to indicate that they are high. Someone going to a party might say "I'm turnt" to indicate that they are excited and looking forward to the party.
This slang has been used by numerous Hip Hop artists, The Dream had a song titled "Turnt" which features Beyonce and 2Chainz. Ciara had a song titled "Super Turnt Up." In Lil Jay's "Turn Up" song, he raps "She Says Molly Makes Her Turn Up - That Goose Makes Her Turn Up - Them Bills Makes Her Turn Up - Imma Throw This Money,Turn Up Turn Down For What."