We have had human level general intelligence for tens of thousands of years and we've not progressed to superhuman general intelligence yet.

General human level intelligence also starts quite low and go quite high, I would say that we're already beyond the lower reaches of general human intelligence.

To say that AGI will instantly transition to ASI is buying into a sci-fi plot or going to beat the early 2000s futurology blogging dead horse where it's assumed that any computer hardware is overpowered and all the magic happens on the algorithm level, so once you crack the code you transition to infinite intelligence overnight, a patently ridiculous scenario where your computer for all intents and purposes casts magical spells(which worked pretty well for the plot of the Metamorphosis of Prime Intellect which I recommend as a read, but it's a plot device, not a realistic scenario)