When children often die from simple dieseases, parents tend to get many kids. When child mortality goes dow so does birth rates.
This has happened almost everywhere so far. Only problem is that there is latency: a period of high birthrate and low mortality before the birthrate goes down. This is what I think is increasing population more than anything else.
In my opinion, we need something like a
Birthright Lottery to keep the population down in the long term, and the sooner we get one, the better. There are other factors than CO2 that need to be controlled.