By getting in the middle of the turning lane (Taking the Lane) you prevent cars from trying to go around you.
When you say that you go through the fence, I can see a path into the Apartment Complex, but that road doesn't lead anywhere. Or do you sneak over to Lakewood Dr SW?
Whether I would choose the left or right lane would likely depend on whether I was headed more to the left or right, and the amount of backed up traffic in the lanes.