In the long campaigns you mention, you have to keep a few things in mind: numbers are often exaggerated, or else include vast numbers of army-following noncombatants. Medieval armies were often followed by huge numbers of camp followers.
We will never really know if they were exaggerated. The information you can dig up today about battle sizes gives varying numbers, but the ranges given are the best guesses of modern historians. I’m willing to go with the opinion of modern historians on this one.
That said, one could argue the numbers in Game of Thrones are exaggerated as well. Most of the information is coming from the characters, and they are not always reliable narrators.
Furthermore, keep in mind that most Medieval battles were not decisive affairs. The two sides came out, yelled a bit, shot a few arrows, maybe have a bit of pushing along a few sections of the line, then, if nobody looked about to break, everyone goes back to camp. These are often nonprofessional armies with limited loyalty to their commanders — decisive field contests of any kind were fairly rare. Sieges likewise were not generally high-casualty affairs; most deaths would have been noncombat deaths too.
Does this contrast with Game of Thrones? We have not seen that many decisive battles in the show (or books), and a lot of the ones we have seen have involved a supernatural element, which totally changes the mathematics.
Interestingly, the Battle of Tours was a decisive affair, while also having relatively low casualties. It just happened that the commander was one of the casualties.
A “long campaign” therefore is a bit of an optical illusion: it could involve steady skirmishing, frequent maneuvers, a few sieges (either completed or abortive), maybe a handful of “battles” with limited casualties… and then in SOME campaigns, you get that rare bird: a decisive, pitched battle.
This is what Robb Stark’s campaign seemed like from the descriptions in the books.
Medieval commanders usually avoided these battles. Armies were nearly impossible to keep organized in the thick of combat. Possibilities for disaster and route abounded. Population growth was very slow, so casualties were hard to replace. Feudal competitors abounded, eager to strike if defeat, or even Pyrrhic victory, should occur. Soldiers could not always be counted on to stay on the right side of battle. Expensive equipment was hard to replace, and trained soldiers took a literal lifetime to replace in the case of mounted knights.
This is where it makes a big difference if you’re talking about the Early or Late Middle Ages. The population of Europe in the Late Middle Ages was double what it was at the time of the Battle of Tours. The growth rate was also double. Soldiers were far better equipped and trained by the Late Middle Ages; there was a far larger professional soldier and knightly class, complete with more sophisticated tactics employed by said trained soldiers that made relatively unskilled soldiers almost worthless. A fine example of this is the Battle of Halidon Hill, where the English professional soldiers obliterated their less skilled and less tactically savvy Scottish opponents.
All of these factors militate towards Medieval commanders avoiding pitched battle. Charles Martels ultimate refusal to avoid combat was a strange strategic turn that took his Arab does by surprise, as was his uniquely loyal, disciplined, and professionalized army. This force does not outlive him, though later Carolingians find substitutes.
Given what I said above, and given the historical inspirations Martin has cited for the story, I don’t think it makes much sense to compare Game of Thrones to the era of the Battle of Tours. Game of Thrones seems to be more accurately compared to the Late Middle Ages than to the Early Middle Ages, and the Battle of Tours was in the 8th century.
Battles in China were often larger, mostly thanks to China’s vastly higher population.
The population of China in the 1200s during the rise of the Mongolian Empire was actually lower than the population in Europe at the same time.
All that said, your overall assertions about Martin and Game of Thrones may still be right, but I remain unconvinced by this particular aspect of your argument.