Never laughed so hard in my life, as I did watching that episode. However, I have since learned that turkeys do fly, at least to short heights. I saw one perched on a street lap post in our community one day and I understand that they regularly perch in trees when they sleep for the night.
*Wild* turkeys can fly. The kind that are farm raised for maximum-breast-meat-size lost that capability long ago.
There are quite a few wild turkeys in my area and I have definitely seen them fly to escape when they thought it necessary. You're right that it tends to be for as short a distance as possible, but it's not necessarily truly short.