Heat the oil in a large skillet over medium heat. Add onion and garlic; sauté 20 minutes or until deep golden, stirring frequently.
Layer turnip slices, onion mixture, and the potato slices in a 13 x 9-inch baking dish coated with cooking spray. Combine the sour cream, salt, pepper, thyme, and broth in a 1-quart glass measure, and stir well. Microwave at high for 5 minutes or until hot. Pour over the vegetables.
Cover and bake at 425° for 30 minutes. Uncover; sprinkle breadcrumbs over casserole. Bake, uncovered, 1 hour or until the vegetables are tender.