I have to confess that coming to the US from Denmark when I came here to go to college, I had heard of turkeys, but never actually seen one. Sure, they roast turkeys in parts of Europe... but not around where I lived. Pheasant and gees? Yes. Not turkeys.
I moved to Texas and there were flocks of "wild" turkeys there... not sure how wild, but we were not far from Mexico. They could fly although they were pretty awkward flyers, but it's pretty impressive (and noise!) to have 20lbs of bird flapping wildly across the neighborhood! Used to scare the bejeezus out of the dogs, too.
That said, I didn't know that all turkeys are from Mexico... learn something new every day!
Yes, they can flap and travel for short distances (like up a fence), but mostly move by walking.