Many developed countries have brought their population growth under control. But developing countries have not.
I'm a social scientist by trade, and the 'problem' you describe is a demographic shift that has been found to happen over and over again.
Prior to the Industrial Revolution, in order to feed yourself when you were old, you needed to have lots of kids to support you. Combine this with high infant mortality rates, and people had lots of kids because they had too. It didn't really matter where you went. Families with 6, 8, 10 kids or more were common. This was not a 'religious' matter; it was a very practical one. And the population didn't grow wildly because the average lifespan was 30s or 40s.
Then the Industrial Revolution came along in what are our considered 'developed' countries like the U.S. We industrialized food production on a scale never been seen in world history, we learned that getting your drinking water from the same location where you dump human waste is a bad idea, and populations skyrocketed. Why? Because people were still accustomed to large families, and average lifespans started increased steadily.
After 2-3 generations of this, more and more people are freed up from agricultural duties, and a great many move to cities for better employment. In cities, having many children is an economic liability rather than a resource, and families scaled back the number of children they had dramatically. Other factors are at work too, but you get the general point.
And now in the U.S., women are only giving birth to 1.9 children, slightly below the number needed to maintain a steady population, which is generally considered to be 2.1 children per female. Among whites, it's only about 1.6. In the U.S. and Europe, total population growth would be zero or even negative if it were not for immigration. In Japan, the low birth rate is only 1.4per female, a number so low that it has many very concerned about the economic condition of the country, especially since the average Japanese person is a ripe 46 years old. By comparison, it's 37.6 in the U.S.
So after nations become 'developed' for a few generations, they almost universally experience dramatic reductions in their birthrates.
For what it's worth, the U.N.'s 'medium' prediction for global populations is that the number will stabilize at around 11 billion or so by 2100. Their 'low prediction is actually that the total population will decrease to about 6 billion. But that doesn't take into account many, many things that could happen one way or the other (i.e. even longer average lifetimes, war, famine, global pandemic, etc.).