Hakurei (aka salad) turnips are small, sweet and tender, but you could substitute small purple-top or white turnips. If you use one of those varieties, peel and thinly slice them.
Heat coconut oil in a large skillet over medium heat. Add ginger and garlic; cook, stirring constantly, until soft and just starting to brown, about 3 minutes. Add bok choy (in batches if necessary), turnips and chile. Add water and salt. Cook, stirring frequently, until the bok choy and turnips are tender, 5 to 7 minutes. Drizzle with sesame oil before serving.