If using wood skewers, soak in water for at least an hour before using.
To make the turnips: Bring a pot of water to a boil over high heat, add the turnips, and cook for 2 minutes. Drain, add to a bowl, and toss with the olive oil, salt, and pepper.
Preheat the grill to medium-low. Thread the turnips onto skewers and grill until lightly charred, 2 to 3 minutes on each side.
To make the dill oil: Combine all the ingredients in a bowl and whisk together. Drizzle the dill oil over the grilled turnips and serve while still hot.