It doesn't just happen when both people are trying to turn. I have a busy multi-lane road near me with a left turn lane that's just past a stoplight (the stoplight is behind me, but in front of the people whose lanes I need to cross to get to where I need to go.

I'll be stopped in the turn lane waiting for traffic to clear. The stoplight will turn red, and oncoming traffic will back up past the turn lane I'm in. Someone in one of the oncoming traffic lanes will stop before the turn lane to try to let me past, but the people in the other oncoming traffic lane (who are now in my blind spot thanks to the car that stopped to try to let me through) will keep on coming. That's just an accident waiting to happen at that point.