Watch for the green light while it is red or not, and count how many cars are ahead of you waiting to make a left turn. If there are none or just one, your wait time between lights is non-existence. You will be turning left at the next green light because on an average two cars turn left per green light in Los Angeles traffic.
Pull forward into the intersection when the light turns green if you are one of the first two cars. Los Angeles drivers anticipate you to do this or you will just get honked at repeatedly.
Turn left the moment the light turns red. If you are the first of the two cars in the intersection or the second car, stay cognizant of pedestrians and cross traffic, but make the turn fast. Sounds strange, yes, but it is fine to turn on a red light. The oncoming traffic anticipates it. Just move as quickly as possible to get out of the intersection.
Think in terms of patience, reaction time and yellow and red lights when driving in LA traffic. When the light turns yellow don't run yet, cars from opposite direction are still getting through and faster. Be ready for your left turn at the red light, as long as cross traffic has cleared.