brined turkey and fat
Brining shouldn't have any real effect on fat content, just moisture. The issue is, that fat and water can have similar cooling feelings on the tongue, and dried fat may not feel as greasy to the hand.

I have noticed that if you take the bird straight from the brine to the pan, the skin doesn't crisp as much, leaving it feeling mushy and greasy. The best thing you can do here is after you're done brining it, dry it off thoroughly, and have it rest in the refrigerator for 3 or more hours. This will dry out the skin, and make it crispier, and the subcutaneous fat should effectively fry it.

Alternatively, you could have just gotten an unusually fatty bird. It's hard to say from just one sample.