Are your dash light off also? Is that circuit blowing fuses? If your not blowing fuses, then your turn signal/headlight switch is going bad. It cost abour $60 for a new one which is a sub unit of the column levers used to activate light/wipers/lights/headlights. Don't let them sell you the whole unit, it cost about $300.
Fuse panel.
Did you check the fuse box? Sounds like a blown fuse to me.
Short in the wiring or the lamp holder for the tail light is corroded.
A short in the tail light circuit. Brake and turn are on another circuit. Dash light power is fed from the tail light fuse to the dimmer switch and on to the dash lights. Look for a short in the tail light wiring circuit. Also check the license plate light for a short as it is on the tail light circuit also. Had a car with blown fuse, traced the circuit for tail lights and found nothing. Jumped the fuse and burned out the wiring to the tail lights. Come to find out, the short was in the license plate light. Also a bad headlight switch can cause this. Both tail light circuit and headlight circuit go through the switch. Contacts for the tail lights may be bad (defective switch common).
Tail light assembly short
There is an over-current condition in the tail light circuit. It could be a short to ground or a bad bulb.
I think its on the driver side there should be a panel on the right hand side near the excellerator
The tail light fuse is located in the fuse box which is located in the glove compartment. The tail light fuse will be a red wire.
The dash light fuse gets its power from the tail light fuse. If the tail light fuse blows, you won't have any dash lights either.
The dash lights and the tail lights are on the same fuse. In all probability, you have a short circuit in the tail light wiring.
The 2007 Chevrolet Malibu tail light fuse can be found in the fuse box. The tail light fuse will be in the second column, third from the top.
Short in the wiring or one of the tail light sockets is corroded or shorted out.