What? No one said anything about “crossing over a lane of traffic”. You start in front of the cars in the regular go-straight lane, and then when the light turns green, you ride straight across the intersection. If there’s a bike lane on the other side of the intersection, you aim for the bike lane.
This is safer for everyone involved than your suggestion of parking your bike in the turn lane.
This is safer for everyone involved than your suggestion of parking your bike in the turn lane.