Remove the greens from the turnips and reserve. Peel and cut the bulbs into 1/4-inch wedges. Remove and wash the tender leaves from the stems of the turnip greens. Chop the leaves and set aside.
Heat the butter and oil in a large cast-iron or other heavy-bottomed large skillet over medium heat. Add the turnip wedges and sprinkle with salt and pepper, stirring to coat with the butter and oil. Cook, stirring and flipping occasionally, until they are caramelized and tender, about 5 minutes per side. Add the honey, vinegar and pomegranate seeds, then cook until the mixture is combined and syrupy, 2 to 3 minutes. Add the turnip leaves and saute until they are wilted and darken in color, another 2 to 3 minutes. Transfer to a serving dish and serve immediately.