“That’s not what I saw. Stewart had no problem with Barton’s comebacks.”
Also watch the first 9 or so minutes in the Full Episode section. But yes, he did have a problem, he even mockingly complained about his flawed staff “research” cards toward the end. He tried several times to catch Barton contradicting himself when he read various quotes and written phrases of DB, but Barton clarified it every time (essentially correcting him) and took the wind out of Stewart’s attempted gotchas.
I would return to the video and watch what you are describing but I cannot stand Jon Stewart, even though I might side with him in this interview. Stewart and Colbert are pathetic.