Turkeys are native to the continental USA. I suppose they would not have moved as deep into Minnesota if not introduced or possibly reintroduced by man.
Turkeys can be intrusive to ranch yards and can be destructive to stacks of hay bales, but not as intrusive as deer.
Are turkeys any more invasive than smallmouth bass, muskie, or even walleye in lakes where it was man that put them there and not nature? or lakes that lack natural reproduction of the stocked fish so any long-term sustainable population has to be replenished (restocked) ???