My oldest started school reading chapter books. At 5 he was also a summer birthday boy so expectations were not high. He blew them away with how quickly and easily he learned. So in K they taught him to play chess. In second grade he was formally identified because he was reading at least 2 years ahead and was at least 1 year ahead in math (that's school criteria).
My younger son was identified in first. Also several years ahead in reading and math. I am curious whether they are just bright students or Gallo to gifted. My boys are opposites in every way except are both exceptional students and are both left handed. It's interesting. If I ask for IQ tests what reason do you give?
You might like this article. It shows generalized behavioral differences between high achievers (good students) and high intelligence.
http://www.bertiekingore.com/high-gt-create.htm. In general, children who have "enriched environment" in preschool years - that is to say children who are advanced because they had a head start being taught academics - will "regress to the mean" by 2nd grade. That's why so many gifted programs in schools don't start until 2nd or 3rd grades! They're trying to prevent accelerating children who will not be able to maintain advancement.
If your children are past 2nd grade, more than a year advanced, and the school has already identified them as gifted, I think you can trust that they are advanced. If your children's intellectual needs are met with their current program, no further testing needs to be done.
It's quite common for parents to experience denial about their child's abilities! I am just beginning to be able to admit to myself how advanced my son is. Even on this board I often underestimate or understate. I am stuck in my mind believeing initial (incorrect) testing and saying he is "moderately gifted" when he's actually "extremely gifted". I have trouble seeing him for who he is and not what the tests have said. I appreciate how IQ testing gives me confidence in the schooling decisions I've made, but I don't like how much it puts people into categories and how easily I get stuck thinking about how people within a category "should" be. I get stuck thinking "extremely gifted children should have XYZ trait" and when I don't see it in my son I get a little insecure. Had I not tested, I wouldn't struggle internally. However, had I not had him tested, his ADHD wouldn't have been diagnosed and he wouldn't have appropriately challenging schoolwork. The benefit outweighs the negatives for me, but I always encourage people to think about what IQ testing might achieve for their families before requesting it.
I am not familiar with requesting IQ testing from a school district. The first time I had my son's private IQ testing done, I stated the reason was investigating if acceleration might be appropriate. The second time, I stated the reason was to see if his results aligned with his school acheivement. At the time I was concerned I was pushing him too much academically. I think the reason statements on the forms just guide them for what tests to administer.