You looking at the phenomenon of demographic transition. In per-industrial societies population growth tend to be exponential stopped only by hard resource limits in what is known as the Malthusian crisis. Both birth and death rates were very high.

Since industrialization however fertility rates eventually declined along with increase in income per capita. Higher income per capita tended to be linked with increased access to contraception, greater status, rights and education for women, and changing cultural values which stop favoring larger families. This trend has being observed in just about every single country and culture across the globe for the last 50 years.