In a large soup pot, or Dutch oven, combine the leeks, garlic, carrots, celery, turnips, potatoes, bouquet garnish, 1-1/2 quarts water, 2 to 3 teaspoons salt and pepper, or to taste.
Bring to a boil; reduce heat; cover and simmer 40 to 45 minutes, until the vegetables are very soft.
Pass the soup through the coarse blade of a food mill, or purée using a blender or an immersion blender.
Return soup to the pot and whisk in 1/4 cup crème fraîche, or more, to taste.
Heat through, taste and adjust seasonings; be generous with salt and pepper.
To serve, garnish each bowl with a spoonful of crème fraîche and a sprinkle of parsley or tarragon.