But do we have enough time?

Personally, I think that many countries are learning from their overpopulation mistakes. China, over time, has finally started to see a reduction in population growth, and I think India will eventually learn too. Japan has an opposite problem, and although they still aren't accepting immigrants, they have started to promote families with more than one child, simply because they are realizing the aging of their population and the future shortage of young adults in the the work force. Many women today in countries like India and China are being educated to a much greater level than ever before, although not nearly as much as women in the United States, but the male-child preference certainly is dropping to some extent as younger generations have more modern views of their old-fashioned cultures and societies. Education regarding contraceptives has spread worldwide, but there is still much more we can do.
Overall, I think that, if humanity was given enough time to correct the problem, the population explosion would correct itself. That's not to say we shouldn't be doing all we can to resolve the problem now through education, employment opportunities for women, missionary trips, etc. But humanity has overcome two world wars, the bubonic plague, the Great Depression, and so much more. I believe that everything happens for a reason, and you should always learn from your mistakes. However, in our case, there may not be time to learn from this mistake because the problem is so enormous to begin with.
If a fire was approaching a village, and only those on a hill high above the village saw the fire, they would act immediately to save their homes, even if they were the only ones doing so. Eventually though, those in the very heart of the town would see the fire too, but by then it would most likely be too late. I think the same thing has happened to us. If we were given the time, ability, and resources to change, global population growth would decrease. However, this planet is running out of natural and nonrenewable resources. No environment can support infinite population growth and expansion, and Earth is no exception. There is no way to know for sure what the outcome of such a situation would be because one has not happened yet. There is no way of knowing whether the problem would fix itself or we would all die from the exhaustion of Earth's natural resources. The only sure way to know the exact end result is to live through it and watch it happen firsthand....