Strangely it's kind of the other way around. When you have a poor, unhealthy population with labor-intensive economy, high child mortality and low life expectancy, you tend to get tons of people.
Why? Because it's economically smart to have children.
In a labor-intensive economy, you can generate more income in the family with more children.
And in a poor subsistence economy, there is no surplus wealth being generated to go to the state for redistribution and another thing I'll get to in a minute. This means that institutionally-provided social security is often non-existent (no pensions, no insurance, no universal healthcare, no benefits, food stamps etc). So instead of institutions, these services are provided by the community, i.e. children. If you have 5 kids, there's a much better chance of someone taking care of you at old age, or when you're sick, than if you have few or no kids.
You have more kids because there is a higher chance of death. Child mortality under five is absolutely STAGGERING. I literally think about this every week and wonder why we can't get the news to report on it on a weekly basis. The number is about 17 thousand per day. Try putting that into perspective. That's more than 5 times 9/11 per day, or 50% the death-rate in World War 2 year in year out. At these rates, it makes sense to have more children, parents actually expect not all their children to last.
When you take death and sickness out of a community however, there is much more chance of economic growth. There are loads of UN reports on how mortality and sickness affects education, employment, parenting, productivity etc. In short, if you improve these things significantly in a country, 10-20 years later you see millions of extra people pour into the middle class.
And what happens then? The truth is that it's economically disadvantageous to have children in a wealthy society. Children cost (way) more money than they generate. Wealthy societies have child-labor laws (well, every country does, but they're enforced more strictly), as well as laws that require all children to go to school. Standards of living in a wealthy society requires loads of expenses on children, and price levels aren't low either. And lastly we see that in wealthy societies that there is more wealth-surplus, leading to cultures that have the time and literally the money to explore non-economic interests, like the arts, and this often goes hand in hand with less time spent on simply surviving and taking care of family members, in fact leading to sharply reduced birthrates as new generations grow up with a bigger interest in traveling than raising a family. (extensive traveling for recreation being something almost only possible in wealthy societies with a wealth surplus, i.e. the minimum amount of work you need to do just to survive is something like 30% instead of 90%, so if you spend 60% of your time working you have a large wealth surplus and a lot of free time).
In short, having 10 children is the 'worst idea ever' for most parents both economically and culturally the richer a country gets.
What does this all mean? Well, if you believe me, we've established that it makes sense to have lots of children in a poor country, and very few in a rich country, which is easily seen in empirical data. (it's not a cultural thing, most western countries with low birthrates had very high birthrates before they became wealthy post industrial revolution). And we've also established that mortality and sickness are huge inhibitors of economic growth.
Lastly, I'd say that a richer society with less mortality and sickness is more stable, better educated, has better funded institutions, which means natural family planning (like contraception and sex education in best case, or authoritarian China-style 1-child policies in worst case) should also be much more effective.
In other words, in the long run, reducing death rates is actually likely to decrease the size of the population, with of course the added benefit that it's freaking awesome to save lives that every living person deserves to have. After all, none of us would volunteer to die just to be able to 'sustain the planet with a lower population'! :) But I tried to ignore this point for argument's sake.
The corollary to this story is that a wealthier society consumes much more, which is less sustainable. (e.g. 500 million people in China/India who join the middle class and turn to 'basic' items like toilet paper is like adding another Europe or US. It's insane.) The fun and optimistic corollary to that is adding 500 million educated brains, who knows how many nobel prize winners will be among them providing the solutions we need :)