I think this what's happening to your tank.

I don't think it's a battery discharge problem because you can move forward and back fine right?

I think it's your motors that are drawing too much current and the board just shuts down. The reason why it only happens when you turn is because one motor is basically moving the entire tank weight, so that motor is drawing much more current than when two motors are working together to move the tank. The reason why the right is more affected than the left is because your your left track is more rigid, it takes more torque to move it, hence more current being drawn by the left motor.

By any chance you're using metal tracks too? Metal tracks requires more torque to move, so taking them off might help.
Do the following test: Take off the tracks and test your motors just with the sprockets. If they run fine without the load, then it's the motors drawing too much amps than your ESC can handle.