Cook sausage in a skillet or frying pan over medium heat, rotating every 3 minutes, until nearly cooked through, about 10 minutes. Drain most of the juice, leaving a layer of juices in the skillet.
Stir butter and olive oil into sausage; add turnip greens and garlic. Cook greens mixture over high heat, stirring every 5 minutes, until wilted and sausage is cooked through, about 15 minutes. Season greens mixture with salt and pepper and continue cooking over medium-low heat for 10 minutes more.