The ancient underground cities of Turkey are dug into soft, dry ash above deep streams. Are there similar locations in North America that such structures could be dug/constructed?
Yes, lots of places. The USA happens to host the largest supervolcano, Yellowstone, and every half million years (or so) it covers the US in a layer of ash. There are plenty of locations around Nebraska or Wyoming with deep ashbeds. . At its deepest it is about 200m thick, but generally it is much less than that, but still plenty thick enough for caves to be dug, if you are so inclined.
By Jim Peaco, NPS - Yellowstone's Photo Collection, Public Domain,
In the image you may see the "Lava creek tuff", ash from the most recent Yellowstone eruption (the pale band on the cliff face). There are streams in the valleys.
More generally the conditions for producing layers of volcanic tuff is not exceptional and can occur wherever there are volcanos that produce large "Plinian" eruptions. And there are plenty of other rocks that are soft enough to be dug with simple hand tools: Chalk, shale, mudstone and others.