Preheat oven to 400 degrees then coat a rimmed baking sheet with cooking spray. Scrub beets, turnips, carrots and potatoes under cool running water then blot dry with a kitchen towel.
Trim the root and stem end from beets then place each beet on a separate sheet of aluminum foil just large enough to wrap them. Drizzle each beet with a small amount of olive oil then sprinkle with salt. Wrap them tightly in foil and set aside. Repeat this process with the peeled garlic.
Peel and trim turnips and carrots. Cut turnips into 1½" cubes and the carrots into large, equal-sized chunks. Cut potatoes in half lengthwise then chop onion into 1½" chunks as well.
Combine turnips, carrots, potatoes, onion and rosemary sprigs into a large bowl then add 3 Tbsp olive oil. Sprinkle with kosher salt then toss until completely coated. Transfer vegetables onto prepared baking sheet, spreading them into an even layer. Place the wrapped beets onto the baking sheet then place in the oven for 45–55 minutes turning every 15 minutes.
Remove veggies from oven. Set beets aside then transfer remaining veggies to a serving bowl. Unwrap beets and place them on a non-porous cutting surface. Use a paper towel to rub the skins off of the beets and discard. You may want to use vinyl gloves to avoid staining your hands. Cut beets into large chunks then add to the serving dish. Season to taste with additional salt and pepper.
Leaving the rosemary whole is a good way flavor the dish without overpowering the taste of the veggies.