The population really is that small. In the first chapter of The Hunger Games, Katniss describes the population of District 12 at the reaping:
The square’s quite large, but not enough to hold District 12’s population of about eight thousand. Latecomers are directed to the adjacent streets, where they can watch the event on screens as it’s televised live by the state.
Recall that anybody between the ages of 12 and 18 are eligible for the Reaping (the event where the tributes are chosen). About a thousand or so 12-18 year olds in a population of eight thousand doesn’t seem unreasonable.
(This passage also explains why we don’t see eight thousand people in the square in the film.)
We know that Panem is based in North America, and the 2009 US Census pegs the number of 10-19 year olds at ~13%. That’s not an exact match, but it does suggest your estimate of a thousand children is credible (as 13% of 8000 is ~1000).
That said, working off these numbers, 1000 is probably a lower bound. In Chapter 9, Katniss hints that the population of District 12 is skewed towards the young:
In District 12, looking old is something of an achievement since so many people die early. You see an elderly person you want to congratulate them on their longevity, ask the secret of survival.
So our estimate of ~13% is probably a little low. Even so, I don’t think a thousand or two is an unreasonable number of children at the Reaping.