Put the turnips in a steamer rack over boiling water, cover, and steam until they're tender enought to be pierced with the tip of a knife, about 10 minutes.
In a medium-large pot, bring 2 quarts of water and 1 Tbs. kosher salt to a boil. Fill a large work bowl with ice and cover the ice with cold water. When the salted water is boiling, drop the fava beans into the pot and cook for 2 to 3 minutes, until the skins are loosened and the beans are just beginning to get tender. Using a mesh strainer, transfer the fava beans to the ice bath. Let cool for one or two minutes, then transfer the beans to a dishtowel or paper towels to drain. Repeat the same method to parboil the peas. When fava beans are drained, slip off their skins by pinching each bean at one end with a fingernail. Set the peas and fava beans aside.
Cutt the whites and tender greens of the leeks into 2-inch lengths; you should have about 16 pieces. Cut enough of the remaining tender green tops into 1/2-inch-wide slices to measure about 1/2 cup. Set the leeks and greens aside.
In a large casserole or soup pot, melt the butter over medium-high heat until foamy. Add the shallots and sauté until lightly caramelized, about 10 minutes. Pour in the white wine and, using a wooden spoon, scrape up any browned bits from the bottom of the pan. Add the stock, potatoes, 1 tsp. salt, and pepper to taste. Cover with a tight-fitting lid, reduce the heat to medium-low, and simmer for about 8 min. Stir in the carrots and the leekss, and then layer the leek slices on top. Cover and cook until the carrots and potatoes are tender when pierced with the tip of a sharp knife, another 8 to 10 minutes. Add the steamed turnips, peas, and fava beans, turning them gently into the simmering stew. Cook for another 6 or 7 minutes, with the top removed, until the liquid is slightly reduced and all the vegetables are tender.
If you want a thicker stew, scoop out 1 or 2 potatoes, plus a little broth, purée them in a blender, and stir them back in. Adjust salt and pepper to taste, and then stir in all but about 1 tsp. of each of the green herbs. Serve the navarin hot in shallow soup bowls, sprinkled with the remaining herbs.