Preheat the oven to 350 degrees F (175 degrees C). Grease an 8-inch casserole dish.

Place turnip in a large pot and cover with salted water; bring to a boil. Reduce heat to medium-low and simmer until tender, 15 to 20 minutes. Drain and transfer to a bowl.

Mash turnip, 2 tablespoons butter, salt, and pepper together until smooth.

Toss apples with 2 tablespoons brown sugar and cinnamon in a bowl.

Spread 1/2 the turnips in prepared dish; top with 1/2 the apples. Repeat with remaining turnips and apples.

Combine flour, 1/4 cup brown sugar, and 2 tablespoons butter by hand in a bowl until resembles coarse crumbs; sprinkle over casserole.

Bake in the preheated oven until cooked through and bubbling, about 1 hour.