Preheat oven to 425 degrees F (220 degrees C). Grease a rimmed 9x12-inch baking sheet with cooking spray.
Combine carrots, turnip, potato, onion, 1 1/2 tablespoons olive oil, garlic, 1 tablespoon rosemary, 1/2 teaspoon salt, and 1/2 teaspoon pepper; toss until well coated and place on the prepared baking sheet.
Bake in the preheated oven until starting to soften, about 15 minutes.
Remove from oven and place chicken thighs on top of vegetables; sprinkle with 1 tablespoon olive oil, 1/2 teaspoon salt, 1/2 teaspoon pepper, and remaining 1 tablespoon rosemary.
Bake in the preheated oven until chicken is crispy and vegetables are tender, about 15 minutes more.