VULINDLELA, South Africa—Public health leaders in the fight against HIV/AIDS have come to an ominous realization: Progress in cutting new infections has slowed, in part because of a persistent cycle of transmission among young women in sub-Saharan Africa.

New HIV infections have been reduced around the world since the late 1990s by diligent efforts at education, the rollout of antiretroviral drugs and other factors. But declines have lost momentum, for the first time since infections began coming down. Today, the numbers...