Preheat oven to 400°F. Cook turnips in a large saucepan of boiling, salted water until tender, 10-15 minutes. Drain. Return to pan; add cream. Bring to a boil. Reduce heat and simmer until cream coats in warm. Purée turnip mixture in a food processor. Cover and keep warm.
Meanwhile, heat 1 teaspoon oil in a large ovenproof skillet over medium-high heat. Season lamb with salt and pepper. Cook fat side down until brown, 5-8 minutes.
Turn lamb over. Tuck 4 thyme sprigs under; place in oven. Roast until an instant-read thermometer inserted into center registers 125°F for medium-rare, 15-25 minutes. Let rest on a plate for 10 minutes.
Drain fat from skillet; add red wine. Bring to a boil. Reduce heat; simmer, stirring and scraping up browned bits from bottom of skillet, until reduced by half, 5-8 minutes.
Meanwhile, heat remaining 2 teaspoons oil in a large skillet over medium heat. Add garlic and remaining thyme sprig; stir until beginning to brown, about 1 minute; discard garlic and thyme sprig. Add spinach by the handful, tossing and allowing it to wilt slightly between additions. Season with salt and pepper. Cook until tender, about 4 minutes.
Cut lamb between bones into chops and serve on top of spinach, turnip purée, and red wine sauce.