Peel the potatoes, quarter them lengthwise, and cut crosswise into 1/4-inch-thick slices. Rinse well in several changes of cold water and drain.
Cook bacon in large sauce pan over medium heat until crisp. Transfer to paper-towel lined plate and set aside. Remove all but 2 tablespoons of the fat from the pan.
Add butter to keep over medium heat. Add turnips and cook, stirring occasionally until butter is browned and turnips are caramelized. (10-12 minutes).
Add potatoes, broth, and 3/4 t. salt, stir to combine. Cook, covered, over low heat, stirring occasionally, until potatoes fall apart easily when poked with a fork, and all the liquid has been absorbed.(25-30 minutes) Remove pan from heat, remove lid and allow steam to escape for 2 minutes.
Gently mash potatoes and turnips in saucepan with potato masher. fold in evaporated milk, reserved bacon and thyme. Adjust salt and pepper to taste.