Heat 2 tablespoons of the oil in a 4-quart stock pot over medium-high heat. Add and sauté the onion until a light golden colour. Stir in the tomato and garlic. Reduce the heat to medium-low. Cover and continue cooking until the tomato is soft and partially dissolved, about 15 minutes.
Put in the carrots, sweet potato, and turnip. Pour in the water, cover, and bring to a boil over medium-high heat. Reduce the heat and simmer until the vegetables are very tender, about 30 to 40 minutes. Remove the pot from the heat and purée the soup (a hand-blender works well).
Stir in the remaining oil and season with salt and pepper. Bring the soup back to a boil, reduce the heat, and simmer for 2 more minutes, then serve.