Green’s Dictionary of Slang

1. (orig. US) to reject, to snub, to rebuke.

1821		D. Haggart Autobiog. 3: I was sometimes turned down for kipping.
1891		Century Mag. (N.Y.) 6538/2: To turn down, [...] to snub; to suppress [DA].
1903		A.H. Lewis Boss 67: He’s schemin’ to tur-rn Jawn down an’ take th’ wa-ard.

2. (US campus) to become bored, tired.

2016

C. Eble (ed.) UNC-CH Campus Sl. Spring 2016 10: TURN DOWN — become unfocused, tired, bored: ‘Rejection used to make me turn down. Now I just move on’.