Not very high or very far.

Domestic ones may have trouble jus t walking

they could survive if dropped out of a plane. They don't have much wingspan for their weight.

Winner of the Buckeye News Award would tell you no, they cannot.

.... once commonly misbelieved to have been imported from Turkey and hence known as "Turkey bustards", were imported to replace the nearly extinct wild American Bustard as a food source, but were domesticated from the beginning, though some have escaped and interbred. Domestication has produced a flightless bird now described as a separate subspecies from the wild bird.

Interestingly, the bird was domesticated in Pre-Columbian times by natives of Central America, and possibly by a more northern native population as well.

The return of the wild North American turkey from the brink of extinction has been an unheralded success story, preceding the similar recovery efforts made on the part of the California condor.

https://en.wikipedia.org/wiki/Wild_turkey

never seen a flock take off and fly anywhere -- even when chased by a dog. They just run like crazy, disperse, and hop-fly to lower tree branches.

flocks of them take off flying. And plenty that just run when they see something approaching. (upstate NY)

there is a type of thorn apple that has long thorns on it. Pre-Columbus, a thorn was bent in half and put through a kernal of corn, with the thorn having a string made from plant fiber from inner bark, tied to a shrub or tree. A turkey eats the corn, and the thorn pops open, creating something similar to a fish hook that gets stuck in the bird's throat. It sounds a tad cruel, but there were not grocery stores back then.

Interesting story about Jimson weed. Noting that it wasn't easily harmed by cold weather, a farmer tried grafting tomato plants onto a Jimson weed stalk. It seemed to work, and his tomato shoots survived late into the season. But his family landed in the hospital with severe alkaloid poisoning, because the Jimson weed produced the toxins within its roots and leaves, which then infused its way into the tomatoes.

https://archive.org/details/medicaldetective0000roue/page/170/mode/2up?q=tomato

As I noted on another thread, I used to raise turkey for pets. I had a friend who would take a few wild turkey eggs from nests in the spring, and we'd put them in the domestic birds' nests. Both can fly, though domestic ones for only short distances -- it depends upon what you feed them, I suppose, so the ones fattened for market likely never do. I did have one tom that flew out of the fence to "attack" a state trooper's car, due to flashing lights, which was actually funny. Even the trooper thought it was funny.

Wild turkey can fly both higher and for longer distances. A guy where I worked had the windshield on his van smashed by one that flew out of some pine trees along a highway. Wild turkeys have much more dark meat, which is perfect for making pemmican with. It has enough tallow so that you don't need to add anything but dried fruit. We use blueberries primarily, but it is good with any of the wild berries that grow in this area.

I've seen them flap up to a garage roof and such. I've never seen them take a long flight like a hawk or such.

I don't actually know how much escape opportunities they need around here. If you get too close, the male of the group usually starts approaching with a look in his eye that makes you want to apologize and turn around.

Although I don't know if it is because domesticated ones are physically incapable of flying or just don't know how. Keep in mind, we're talking about one of the most stupid creatures on the planet. One that needs to have shiny things put in their feed to attraction them to it so they eat.

I’ve seen that a few times myself, happy dogs 🐶

I personally know someone who works R&D for a glass manufacturer. He gets paid to shoot thawed turkeys, 2x4s etc at windows.

Turkeys fly