I have observed cats around turkeys on a couple of occasions. The cats seemed interested in the turkeys and would do a sneak towards them, but once they got close, it was obvious that they were considering whether they wanted to give it a go or not.
In each case, the turkeys acted only mildly concerned about the cats, and would even approach them with an "if you want some of this, come on and get it" attitude. Eventually the cats backed off. Of course, these were mature birds that were substantially larger than the cats that were eye-balling them as a possible main course for dinner.
I'm not a cat shooter either, but if I saw one taking down a wild turkey while I was out hunting, he would be in big trouble.