By turning on the lights, I'm assuming you mean the exterior lights e.g front sidelights, rear tail lights etc. and that pressing the brake pedal puts out the tail light on one side only?
Do the brake lights work on both sides when the exterior lights are off? If so, both brake light bulbs are ok, as is the brake light switch.
If the problem is confined to one side, then concentrate on this side tail lamp unit.
Try the turn signal on this side with the outside lights off, does the rear (red) light try to flash in sync. with the turn signal?
If it does, this is indicative of a poor ground to this lamp - try a temporary extra ground wire to the lamp or the tail/brake bulb holder (can be done with the lens removed).
Make sure that the tail/brake light bulb is correct for the car, afaik it should be a twin filament bulb of 21/5 watt (i.e. 21w for the brake light and 5w for the tail light. (compare with the bulb from the other tail lamp if this is working ok).
(I don't think your problem is caused by a short to ground, as this would cause a fuse to blow, putting some of the lights out of action.)