This is sort of the case in the US at some crossings. Crossings where there is a separate signal for left turns will turn on the don't-walk sign on that side when the left-turn signal is green.

Occasionally I do see intersections where there is a signal phase where car traffic is red in both directions, and pedestrians get the walk signal in all directions (which is extra handy if you need to go diagonally across the intersection). But these are pretty rare.

Most intersections don't have left-turn signals at all for cars, so yes, there will be pedestrians crossing when cars are turning. And I guess for right turns it's nearly always the case that pedestrians are crossing when cars are turning.

It's not great, but I think there's a trade off to be made. As a frequent pedestrian in a city, I don't want to have to wait until both directions of cars have their turn before I can cross. I get that the current situation does increase the risk to me, but that's a risk I'm willing to take. (But I also jaywalk through don't-walk signs all the time if traffic is clear, so perhaps I'm not the best person to make this judgment.)