Same as in a car. The heavier traffic is, the sooner you have to start moving over left if you hope to get there before the turn. As far as I know, all states allow/expect/require you to be in the left lane to make a left turn.

If traffic is insanely heavy and there's a light, you may have to stay right and use the pedestrian signal. It's not ideal but it may be the best choice in some situations.