Preheat the oven to 400°F. Cut four 12-inch squares of aluminum foil.
Toss the cubed turnips with 1 tablespoon of the olive oil and a pinch of salt.
Divide the turnips evenly among the four foil squares, placing them in the center.
Pat the swordfish steaks dry and season both sides with salt and pepper.
Place a swordfish steak on top of the turnips on each foil square.
Scatter the sliced garlic over the fish and top each steak with 2 lemon slices.
Drizzle each packet with the remaining olive oil and 2 tablespoons of broth.
Fold the foil up and over the fish, sealing the edges tightly to form a closed packet with no gaps for steam to escape.
Place the packets on a baking sheet and bake for 15 to 18 minutes, until the fish flakes easily with a fork and the turnips are tender.
Carefully open the packets, avoiding the escaping steam, and garnish with fresh dill before serving.