Preheat the oven to 425F. Line a sheet pan with parchment paper.
Peel the medium turnips, and cut them into 3 by 1/4-inch sticks. In a large bowl, toss the fries in the oil to coat.
Put the Parmesan cheese, paprika, chili powder, and garlic powder in a separate bowl and stir to combine. Add the oiled fries to the seasoning mixture and toss to coat evenly. Spread the fries on the prepared baking sheet on a single layer.
Bake for 30 minutes or until the fries are brown and crisp around the edges, turning them halfway through for even browning.
Meanwhile, make the dipping sauce by placing all the sauce ingredients in a small bowl and stirring until well combined.
When the fries are golden brown, remove from the oven and serve immediately with the dipping sauce.