Set a large pot over high heat and add the milk, cream, salt, thyme and bay leaves. Peel and cut the turnips into 1-inch chunks. Add to the pot and cover partially with a lid. Bring to a boil, and then reduce the heat to a simmer and cook the turnips for 30 minutes.
Cut the potatoes into 1-inch chunks. After the turnips have cooked for 30 minutes (it takes longer than the potatoes to cook), add the potatoes and continue to simmer until all the vegetables are very tender, about 20 more minutes (the tip of a paring knife should go through with little resistance). Discard the bay leaves and thyme. Drain the turnips and potatoes, reserving 2 cups of cooking liquid, and then mash. Add the cooking liquid and butter. Mix well and season with salt and pepper.