Heat the oil plus about ¼ cup water in a large soup pot. Add the leeks and sauté over medium heat, stirring frequently, until they begin to go limp, about 7 to 8 minutes.
Add the broth, turnips, bay leaves, tomatoes, and mushrooms. Bring to a gentle boil, then cover and simmer for 10 minutes.
After 10 minutes, add a cup of water and return to a simmer. Simmer gently for 10 to 15 minutes longer, or until all the vegetables are tender.
Season to taste with lemon juice, salt, and pepper. Remove from the heat. Allow the soup to stand for an hour or two, or even refrigerate overnight, if time allows.
Before serving, heat through as needed. Stir in the spinach, parsley, and dill. If the vegetables seem crowded, adjust the consistency with more water, then adjust the seasonings and lemon juice. Serve hot.