In tFoH:

He needed the angreal to, set wardings so large, but only just. He had thought that he was strong before, but Asmodean's teachings were making him stronger.

Rand basically says here that what he had thought of as "strong" before was nothing compared to where he was headed.

And this makes sense. We have mistakenly assumed that since Aginor burned out in EotW, Rand was stronger than him. But that makes no sense. Burning out is about not overdrawing. Aginor likely took in more because he was greedy for it, after 3000 years with no saidin, exactly as Ishamael said. It had nothing to do with being weak. On the other hand, Rand was weaker, and felt that even the amount he was drawing was enormous. Since this is the first time he channels consciously, that makes perfect sense. There's nothing in the text indicating he was as strong or stronger than Aginor. All that happened is that they struggled for control over who would draw from the well, and Agnor overdrew in the process.

As for Ishamael in the next three books, there is no reason to believe Rand was as strong as him. It wasn't like Ishy was throwing his full strength at Rand early on. He only did so when Rand was channeling with Callandor.

As for the fight with Asmodean, we have confirmation that sa'angreal increase your strength by a set amount. They not multipliers. Rand and Asmodean seemed equal with the Choedan Kal because they were each drawing exactly half its potential. Rand didn't have to be as strong as Asmodean for that to work.

I know this will be viewed as controversial by some, but at the Eye, I think Rand's effective strength was at Cadsuane's level, no more. By tSR, he had jumped beyond Egwene's (full potential) level, and only after Asmo started teaching him did he jump into Foresaken class. Thus, but the end of tFoH, I'd wager he was somewhere between Asmo and Rahvin. By aCoS, he was at Sammael's strength exactly, and by tPoD, he matches Aginor. I think WH and the Cleansing tipped him to his full potential.