It could be a fuse.
On the trailer, left brake and left turn are the same light. The fact that you get left brake means the bulb and wires to the left light are okay.
If you get left brake but not left turn then there could be a problem on the truck. A lot of SUVs and trucks that came with a "tow package" have a set of relays and fuses specifically for the trailer and it is possible to blow one of those fuses. Truck lights work okay but trailer lights don't.
Before you do anything, however, the first thing I would do is check the ground connection between the truck and trailer. If you don't have a good ground the lights can get screwy.
Does the trailer light connector have a specific ground wire, or does it rely on the hitch for ground? If you have the common four/five pin flat rubber connector then the ground wire it the one odd connector and should have a white wire associated with it.
If you don't have a ground wire then I would add one! Old rusty hitches and/or receivers are notorious for making a bad ground. If you are not sure of the ground grab a pair of jumper cables and connect from truck to trailer frame and try the lights again.
The next thing you should do is test the lights at the truck connector. You can use a voltmeter but a test light is easier. Clip the ground lead to the odd pin (which will be the bare connector on the truck side). Turn the head lights on, test to the next pin over. Turn the left turn signal on, test the second connection over. Turn the right turn signal on, test the 3rd (last) connection.
Here is a good website for a wiring diagram:
http://www.championtrailers.com/4%20pole%20wiring%20dia.pdf If you don't have power at the connector then check the truck wiring and fuses.
If you have power at the connector, and you have good ground to the trailer, the it is either the trailer wiring, a burnt out bulb or the light fixture has gotten flakey.