I'm only a show-watcher, haven't read the books, but...
I thought his reputation was well demonstrated in the scene where he is confronted by Robb after being captured and his proposes to settle the whole war by fighting Robb. Regardless of whether he is actually amazing or not (became the "Kingslayer" by a literal back stab), Robb respected Jamie's rep enough to not fall into that trap.
Also Ned seemed to have his own skepticism about Jamie's opponents, etc.
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