Jaime's skills are being underestimated badly in this thread. He's easily the strongest knight at the start of the Game of Thrones novel up until half way through the Storm of Swords novel and is one of the strongest knights of all time in the history of Westeros.
Jaime said:
"I learned from Ser Arthur Dayne, the Sword of the Morning, who could have slain all five of you with his left hand while he was taking a piss with the right."
Jaime was taught by one of the greatest knights in the history, Ser Arthur Dayne. But even before Jaime was taught by Arthur, he was clearly a prodigy. He won a tourney melee at the age of 13, a 13 year old little boy is beating up a bunch of grown men, that's an insane feat.
Jaime's next crazy feat is being able to cross blades at such a young age with a swordsman as talented and dangerous as the Smiling Knight. Now that's enough of Jaime's hype before the story takes place, let's look at his feats in the actual story because they are even more impressive.
Let's start with Whispering Wood, where Jaime was defeated and captured. We'll start with the obvious fact that it was Tywin's fault that jaime was ambushed and not Jaime's fault. With that out of the way, let's take a closer look at the actual battle, because Jaime was an unstoppable killing machine. Despite being caught off guard due to Tywin's fuck up, and vastly outnumbered, this is what Jaime did:
Glover/Robb/Catelyn/Theon said:
Robb looked away into the woods, with the same brooding look that Ned often got. “He . . . he killed them . . . ” “
Lord Karstark’s sons,” Galbart Glover explained. “Both of them,” said Robb. “Torrhen and Eddard. And Daryn Hornwood as well.” “No one can fault Lannister on his courage,” Glover said. “When he saw that he was lost, he rallied his retainers and fought his way up the valley, hoping to reach Lord Robb and cut him down. And almost did.”
“He mislaid his sword in Eddard Karstark’s neck, after he took Torrhen’s hand off and split Daryn Hornwood’s skull open,” Robb said. “All the time he was shouting for me. If they hadn’t tried to stop him—” “
—I should then be mourning in place of Lord Karstark,” Catelyn said. “Your men did what they were sworn to do, Robb. They died protecting their liege lord. Grieve for them. Honor them for their valor. But not now. You have no time for grief. You may have lopped the head off the snake, but three quarters of the body is still coiled around my father’s castle. We have won a battle, not a war.”
“But such a battle!” said Theon Greyjoy eagerly. “My lady, the realm has not seen such a victory since the Field of Fire. I vow, the Lannisters lost ten men for every one of ours that fell. We’ve taken close to a hundred knights captive, and a dozen lords bannermen. Lord Westerling, Lord Banefort, Ser Garth Greenfield, Lord Estren, Ser Tytos Brax, Mallor the Dornishman . . . and three Lannisters besides Jaime, Lord Tywin’s own nephews, two of his sister’s sons and one of his dead brother’s . . . ”
Theon's quote makes it clear just how outnumbered Jaime's forces were, and he was caught off guard. Fespite those huge setbacks, Jaime Lannister managed to killed Torrhen Karstark, Eddard Karstark, and Daryn Hornwood instantly. And these aren't random fodder, these are highborn men that have been training in the art of swordsmanship and war since they learned how to walk, as is the custom of all highborn men. And Jaime was also about to kill Robb Stark and single handily win the entire war between Houses Stark and Lannister until his sword got stuck.
Jaime almost won a 4 versus 1 fight without getting a scratch and somehow there is a debate about where he ranks in the Westeros hierarchy? Garlen Tyrell's training with two people at the same time looks like child's play compared to this. Robert Baratheon suffered such deep injuries from merely only fighting one man (Rhaegar Targaryen) that he was unable to join Eddard Stark when he initially went to Kings Landing. The Mountain lost his only 1v1 fight that we have seen in the series. Meanwhile literally nobody can touch Jaime or even injure him, he just continues cutting through people like carving a cake. The dude was literally just slicing through Stark men like it was child's play. Robb Stark was so scared of Jaime that he was literally stuttering like a bitch when he was trying to explain to his mom what happened, absolutely embarrassing.
In Clash of Kings, Tyrion comes up with a plan to help Jaime escape, and then we discover that Jaime has killed Poul Pemford, Myles, and injured Delp. Once again, Jaime is taking on multiple opponents and completely embarrassing everyone. This is after spending months as a captive and being unable to train with a sword. Another extremely impressive feat.
In Storm of Swords, Jaime is malnourished, chained at the wrists, and hasn't practiced with a sword for an extremely long time and is able to keep up with Brienne for a while until he gets tired, that's an extremely impressive feat. And what's even more impressive is Brienne's quote after said fight:
Brienne said:
He was weak from imprisonment, and chained at the wrists. No knight in the Seven Kingdoms could have stood against him at his full strength, with no chains to hamper him. Jaime had done many wicked things, but the man could fight! His maiming had been monstrously cruel. It was one thing to slay a lion, another to hack his paw off and leave him broken and bewildered.
The good sis herself says that no knight in the seven kingdoms can stand up to a Prime Jaime. Now one might ask, what makes Brienne a credible source of info? Well for starters, she's a skilled warrior herself. She's also seen countless amount of fighters in her life time, especially since she has participated and ended up winning a tourney melee that featured the best knights from the Reach and Stormlands in a Clash of Kings. There is really no reason to doubt Brienne's words.
But if you still want to doubt me, then let's look at what another credible source has to say about Jaime's combat abilities:
Ser Barristan said:
Black as maester’s ink he was, but fast and strong, the best natural swordsman Selmy had seen since Jaime Lannister.
I really hope that I don't have to explain why someone as old, experienced, and talented as Ser Barristan is a credible source when it comes to swordsmanship? Good.
Jaime Lannister has the best portrayal and feats out of any knight in the history of the Seven Kingdoms. The lion bows to nothing and no one. The author has went out of his way to make this painfully clear on numerous occasions.