Place turnips in a large skillet, then cover them halfway with water. Add the butter, sugar, and a generous pinch of salt. Bring to a boil over medium high heat. Cook, stirring occasionally, until the turnips are tender and the liquid has thickened, about 15 minutes. (If the turnips become tender before the liquid has reduced, transfer the turnips to a dish using a slotted spoon, then continue cooking the liquid to allow it to thicken. Return the turnips to the pan and gently stir them to coat well.)

Place the turnip greens in the skillet over medium heat and cook, stirring occasionally, until wilted, 2-3 minutes. Season to taste with salt, and serve.