Echoing others, wild turkeys can and do fly, when they have to. People in my area have an agreement not to hunt anything in the area, so turkey and deer roam freely, and with no fear of people. Just erect high and sturdy fences around gardens.
My nephews little dog will chase the turkeys when he can, and I’ve seen a few fly over the house. The dog is afraid of the deer, though.
Yes turkeys fly- and they are powerful fliers too- they roost in treetops and last i looked i’ve never seen rope ladders to the tops