Something that works well for a nest for the muscovies, but might work for the turkeys also, is to cut a barrel into fourths. Once crosswise and once lengthwise. You end up with 4 half rounds with a back wall. A board fastened across the front to keep the sides in place and that is it. If placed over dirt, they can dig down which is what the scovies like to do and or fill it with bedding. Size of your barrel will determine the size of the nest.
I may have to try this, my muscovies, are laying eggs out in the goat pen, and well the dogs, casper, and earl, make short work of the eggs. I really don't care if they hatch any, but as hard as they work to build the nests, just might be an option for them.