Preheat oven to 350 degrees. In an 8-inch square baking dish, arrange a single layer of turnips and sweet potatoes, overlapping slightly. Season with salt and pepper and sprinkle with one-third of the flour. Repeat to make three more layers, sprinkling with salt, pepper, and flour between layers. Dot top layer with butter, then slowly pour broth and wine into dish, keeping layers intact. Cover with foil and bake until vegetables are tender when pierced with a knife, 45 minutes.
Increase oven temperature to 425 degrees. Remove foil and sprinkle cheese over dish. Bake until cheese is golden and bubbling, 12 to 15 minutes. Let gratin sit 10 minutes before serving.