I would rule that it wouldn't work that way. Not because the mechanics wouldn't work in principle, but because "turns" are an abstraction; combat actually happens simultaneously. Having darkness "on" during your turn and "off" during your opponent's turn is a deliberate attempt to go against the spirit of how combat works. In actuality, yours and your opponent's turns happen at the same time.
I'd try to find some middle ground because the idea is good, but it certainly wouldn't be as simple as on/off during each turn. That'd be like saying "on my turn, I pop up from behind cover, shoot the enemy, and then duck down again into total cover". RAW that is an illegal move because it only works if you interpret turns as people literally taking turns in combat, which they don't.