I live near a town with what I think is a similar intersection. The law here is that it is always the pedestrian’s “turn” if they choose to enter a crosswalk and there is no traffic light specifically telling them otherwise. In more detail: if the pedestrian wishes to cross at an intersection with a traffic light, they have “right of way” (the legal term that corresponds to your use of “turn”) when the light is green for the lane of travel parallel to their crosswalk; if there is no light, the pedestrian always has “right of way”. For intersections with no light where pedestrians have difficulty crossing the road safety even in the crosswalk, we sometimes set up the warning lights situation. These lights serve the same function as the turn signals on a car: they broadcast a specific kind of intention so others using the road have an easy-to-see and harder-to-ignore signal, and so that everybody involved can feel comfortable that communication has been had.
I live near a town with what I think is a similar intersection. The law here is that it is always the pedestrian’s “turn” if they choose to enter a crosswalk and there is no traffic light specifically telling them otherwise. In more detail: if the pedestrian wishes to cross at an intersection with a traffic light, they have “right of way” (the legal term that corresponds to your use of “turn”) when the light is green for the lane of travel parallel to their crosswalk; if there is no light, the pedestrian always has “right of way”. For intersections with no light where pedestrians have difficulty crossing the road safety even in the crosswalk, we sometimes set up the warning lights situation. These lights serve the same function as the turn signals on a car: they broadcast a specific kind of intention so others using the road have an easy-to-see and harder-to-ignore signal, and so that everybody involved can feel comfortable that communication has been had.