This is a grim time of year for a hospital in the rolling hills of South Africa’s Eastern Cape province, where mattresses are laid on floors to cope with the stream of young men with severe injuries from botched circumcisions at initiation ceremonies. Some patients with amputations, a doctor says, were told by their handlers that their genitals would grow back, reflecting a fog of misinformation that makes it hard to stop what has become an annual health crisis. It wasn’t supposed to be this way. The traditional ritual, which goes back centuries, is meant to usher youths into manhood,...