Heat olive oil in a large saucepan over medium-high heat. Add onion and mushroom; cook until tender, about 3 minutes. Stir in potato, lentils, garlic, parsley, fennel greens, salt, ground black pepper, cayenne pepper, and turmeric. Pour in water; stir to mix. Turn heat to medium-low and simmer until flavors are combined, about 45 minutes.