Heat oil in heavy large pot over medium-low heat. Add leek, celery and garlic and sauté until vegetables begin to soften, about 5 minutes. Add turnips, rutabagas, potatoes, carrots, tomatoes with juices and 2 cans broth. Bring to boil. Reduce heat; cover and simmer until vegetables are very tender, about 45 minutes.
Transfer 4 cups soup to processor. Puree until almost smooth. Return puree to pot. Add remaining 2 cans broth; bring to simmer. Season with salt and pepper. Ladle soup into bowls and serve.