Jordan Baker's recollection of Gatsby's past seems to be much more direct and honest than Gatsby's retelling of his history.
At the end of Chapter 4, The Great Gatsby shifts away from Nick's narration to that of Jordan. This is an interesting choice in that it gives more gravitas to Jordan's statements about how the relationship with Jay affected Daisy. In her words, Jordan seems to admire Daisy in much the same way Nick admires Gatsby ("I was flattered that she wanted to speak to me, because of all the older girls I admired her the most."). In addition, Jordan is reporting actual events about Daisy, especially on the day of Daisy's wedding.
In addition, there are no contradictions in Jordan's story, while Gatsby's stories about himself are full of contradictions and half-truths. There are certain aspects of Gatsby's history that are undoubtedly true: he went to war and "tried very hard to die," but ended up a war hero honored by "little Montenegro," a claim supported by an actual war medal. However, stories about him traveling as a "young rajah" and being "educated at Oxford" seem far-fetched, although he does have a photograph of himself at the English university. Then there are outright falsehoods. Gatsby says that he came from "wealthy people in the Middle West" and then goes on to say that "Middle West" means "San Francisco."
Is there truth in Gatsby's story? Absolutely. But the lies cloud the truth. In Jordan's story, there is no motive to lie. So her story is much more believable.