According to the Emory School of Medicine, IQ stabilizes around age 4 for most children (later for those who were born premature or had other significant health issues).[1]
Because they have been doing such testing and assessing for longer than muliple entire human life times, and have had time to see the full and complete eventual results of past procedures and conjectures, good and bad.
Either early bloomers are not a thing, or they are accounted for and the tests and assessment are not as simple as you aparently imagine.
The question already granted "assume the testing and assessing is all done correctly" and only asks how they know something doesn't simply look misleading because some kids might just reach their final plateau earlier than others.
That is very simply answered by simple time and numbers, once enough time has passed to see a large number of entire human lives while under any sort of methodical recording. That has happened by now. We have data on a large number of entire human lives.
It's only an appeal to authority in so far as I do assume that the people whos job is to collect and reason about that data have done even the most cursory job of it.
Saying that a certain property doesn't change much for most people after a certain age is not something that is likely to be hidden or hidable in the data. It sounds quite basic to me, and sounds like something anyone would see and no one could probably argue with. It sounds like the kind of fundamental and obvious thing that I assume if anyone did try to misrepresent it, too many other people would speak up.
I am only trusting that there isn't some kind of massive conspiracy that every nurse and med student have somehow all agreed to keep this ond wierd thing a secret for some reason.