Peel the turnips and cut them in half lengthwise. Cut each half into wedges 1 inch thick at the widest point.
Put the bacon pieces in a large skillet and set over medium heat. Cook, stirring occasionally, until crisp, about 8 minutes. Transfer to a plate lined with paper towels. Pour off the bacon fat from the pan and set the pan back on the burner. Add the butter and, when it’s melted, add the turnips, apple cider, sugar, and 1-1/2 tsp. salt. Bring to a boil over high heat. Cover the pan with the lid slightly askew, reduce the heat to medium high, and cook at a steady boil, shaking the pan occasionally, until the turnips are just tender but not soft (a pairing knife should enter a turnip with just a little resistance), 8 to 10 minutes. Meanwhile, peel and core the apple and cut it into 1/2-inch slices.
Uncover the pan, add the sliced apples and sage, and continue to boil, stirring occasionally, until the liquid has reduced to a sticky glaze, 2 to 3 minutes. The turnips should be soft and the apples should be crisp-tender. (If not, add a few tablespoons of water and continue to cook for another 1 to 2 minutes.) Toss in the cooked bacon and season to taste with black pepper and more salt if necessary.