SEOUL — A crowd of young women wait nervously in the lobby of a popular plastic surgery clinic in Apgujeong, the affluent neighborhood at the heart of Gangnam. Photographs of Korean pop singers and actresses line the walls, winsome customers who smile next to their cosmetic surgeons. “It’s painful, but I really want a face like those Korean actress girls,” says a Chinese patient leaving a check-up — with her nose wrapped in a surgical bandage.