Where education has been available to girls for a few generations, where people have, on average, enough to live on reasonably comfortably, where there is at least some free health care, where there is clean water and sanitation, where, in short, living standards have risen and people have more 'security' and don't need to rely so much on their children as they grow old, population growth rates have fallen naturally. Most European states are a classic example. In France they fell so much the government actually encourages larger families.
Seems to me that the solution, though slow and complex and difficult, is obvious. Educate women and encourage prosperity for "the masses".