In A Game of Thrones, all we have to go on is hearsay. Tales of how impressive the Khal was in battle, the fact that he has a massively long ponytail suggests he was never defeated in battle, the fact that everyone fears and respects him. But we never see first hand evidence of his skills, strength and endurance. He has a fantastic mystique and aura of death about him, but it's all just that: mystique and aura. This was somewhat remedied by the TV show, but I maintain what I thought originally: that wasn't a fight, that was an execution. Drogo was barely scratched by the "challenger," and he punished the would-be usurper with contemptuous ease. Aside from that, we have practically no information to go on.
Conan, of course, is a different story. We have plenty of first-hand accounts of the Cimmerian's battle against scores of different opponents. Duels, brawls, skirmishes, assaults, battles, sieges, with odds both against him and in his favour. So we have a good idea of how Conan would fare against, say, Thomas Covenant, Imaro or Elric.* But against Drogo, all we can do is speculate. All the more reason to be annoyed at Martin killing off one of the best characters in A Game of Thrones.
Conan, being a rather famous example of the badass barbarian warrior, is a regular combatant in such contests. However, given how vastly different Howard's Conan is from film-Conan, who is different from Marvel Conan, who is different from animated Conan... So you can have matchups go all over the place. And, of course, all are decided by vote.
Sometimes he's really punching above his weight:
|Loss||Witch-King of Angmar||56||to||62|
Others, he's come out on top when circumstances really point against it:
And, predictably, there are other battles he really should've won:
That said, when it comes to fictional characters, the only person who would really win is who you want to win: all the attempted empirical deduction and analysis is, unfortunately for such pedantic folks as myself, subject entirely to the whim of circumstance, bias and story.
*Effortlessly, Decently and Poorly respectively.