There's a famous episode from the old TV sitcom "WKRP in Cincinatti" in which the station's uber-dweeby weatherman Les Nessman arranges for live turkeys to be dropped from a helicopter for a Thanksgiving stunt. Of course, it turns into a bloody debacle as they crash en masse on the pavement in front of horrified onlookers. Les's memorable line: "As God is my witness, I thought turkeys could fly!"
I don't know about farm turkeys bred for the table, but I can tell you that wild turkeys can fly. I've seen it. It's not pretty or graceful, it's loud flapping in an awkward fury, but as an escape device, it does the job.
I was out in the pines walking the dogs when I heard the commotion. Lena, the hyperactive GSP, had scared up a wild turkey. I looked and saw the big tom pounding the air with its wings and rising with weighty insistence up and over the tops of the (easily) 40-foot loblollies. I had in the past seen one fly up to a tree branch maybe 20 feet off the ground, but this was more impressive. The canopy obscured its path and I wasn't able to follow it, so I don't know how far it flew. I just knew it successfully evaded the fastest, most energetic and pound-for-pound strongest dog I know.
As God is my witness, I know turkeys can fly.