Peak season around the Mediterranean may be winding down, but fall and spring are when the region is at its best: The sun still shines, the crowds thin and prices drop. For your next shoulder-season getaway, head east past the usual beach clubs of St. Tropez and Positano and you’ll reach Bodrum, a peninsula on Turkey’s Turquoise Coast that has undergone a remarkable transformation. When I first visited in 2012, the number of bland all-inclusive resorts and tchotchke-filled bazaars suggested that the former fishing village and artist colony had fallen victim to a familiar Mediterranean trajectory, from idyllic escape to…