All three areas seemed to be equally fertile with plenty of farming potential, so how come India and China accelerated ahead in population so early?
There are a couple more important reasons.
First, they have very ancient civilizations. Civilizations are generally organized, which means they are better able to deliver resources to support relatively large populations. For example, large organized labor pools for public works projects to create canals for irrigation or to bring water to the cities, roads for transport, large scale trade, both internal and external.
Second, they had a lot of land suitable for growing food. In order to support a large number of people, you need a correspondingly large amount of calories. If you rank countries in terms of the area of arable land available for cultivation, the ranking is:
| Country | Arable Land (sq km) |
|---|---|
| US | 1,650,062 |
| India | 1,452,076 |
| China | 1,385,905 |
| Russia | 1,174,284 |
| Brazil | 586,036 |
| Canada | 415,573 |
You can see that India and China both have huge amounts of arable land. Only the US has more, but the US didn't historically have agricultural civilizations across this land, much of it being occupied by hunter-gatherers.
In fact, India actually has more arable land than the US, if you count only land that is suitable for natural irrigation (not including modern irrigation such as tube wells run on electricity or diesel). This is surprising, seeing that India is much smaller in terms of area than huge countries like Russia or China or Canada, but such is the case.
Rich agricultural land that is cultivated by organized civilizations is very productive, and the enormous amount of calories it produces can sustain very large populations.
For most of history, to the best we can calculate such things, India and China have each accounted for about 25% of the world's population, and they do so to this day.
Europe has much less agricultural land than India or China (even if you add up all the countries). Also, the land is not as productive because the growing season is much shorter in Europe, and gets even shorter the farther north you go. India and southern China have tropical climates, which allow multiple growing seasons every year.