Birth rates are falling across the board, but stopping it faster either requires severely authoritarian measures or increasing economic safety, which counters the benefits. And reducing birth rates too fast sets us up for disaster too.
All it requires is education. I'd say a quarter of the over educated I know do not have kids, myself included. After a certain level of education, our society strikes one as no place moral to bring more people.
To allow education of enough people requires a significant economic lift. Not going to happen fast enough.
UN projections puts the end of population growth at some point within the next century, but accelerating that change fast enough to have an impact on climate change would require dramatic interventions, and also will not happen because it would be absolutely politically untenable for most politicians to stunt growth that way (you'll note, politicians many places are encouraging more children, and China has kept loosening up their own policies because current birth rates are causing concerns - politicians would in general rather encourage more children than open up for more immigration, and opening up for more immigration is in any case a stopgap).
So, while reducing growth is absolutely part of it, we're already doing that pretty much at the rate we can expect to be politically tenable. There are parts of Africa we could encourage a faster reduction in (e.g. some of the absolute worst, like Niger, would likely fall off a lot faster with more economic development assistance), but it's not going to do enough to speed up population reversal in time.
Remember that changes in birth rates has a trailing effect of a couple of decades, typically. So even if we could make changes now to education for example that will drastically cut birth rates eventually, it'd not affect population size enough in time.
