It's possible that Jim Bridger experiences more growth than any other character in The Revenant. After all, Hugh Glass starts as the toughest man of all time and ends as the toughest man of all time—not much evolution there. Young Jim Bridger, on the other hand, grown from a boy to a man.
Before we dive in, it's important to establish why Bridger is with the Rocky Mountain Fur Company in the first place. Bridger has been obsessed with the idea of exploration since an early age, and although a traumatic early life prevents him from fulfilling those dreams, he eventually makes his way to St. Louis, where he gets the job. This is an important moment: he had come "to believe that going west was [...] a part of his soul, a missing ounce that could only be made whole on some far-off mountain or plain" (1.6.34).
Unfortunately, when we first meet Bridger, the kid is too insecure to follow his dreams to the fullest of his abilities. We see this lack of confidence most prominently in his interactions with Fitzgerald: although Bridger knows that caring for Glass is the right thing to do, he allows himself to be controlled by the older man. Ultimately, this insecurity leads Bridger to follow Fitzgerald's lead as they steal Glass's belongings and abandon their comrade for dead.
But then something amazing happens—Bridger grows a backbone. We see a hint of this when Bridger, finally fed up with Fitzgerald's insults, points a gun at his tormentor's head. That gets the message across. Later, we see Bridger's growth emphasized further when he takes over navigational duties from Fitzgerald as they travel to Fort Union—he had to do this, given that Fitzgerald can't navigate worth a hoot. In fact, this experience shows Bridger just how skilled he is compared to most of the punks who end up on the frontier.
Even with this confidence boost, however, Bridger still has a big problem—in the form of Glass. In contrast to Fitzgerald, Bridger has felt guilty about the abandonment since it happened, and this guilt has been tormenting him in the form of strange nightmares.
As it happens, Bridger's guilt just happens to save his life. When Glass finally finds and attacks the kid, he looks down into Bridger's eyes and sees "not malice, but fear; not resistance but resignation" (2.21.92). In other words, Glass sees Bridger's shame. He's utterly shocked by this and finds himself forgiving the kid he swore to kill. Maybe he also understands that Fitzgerald has put Bridger up to everything, and the Fitzgerald is the one he really needs to wreak his vengeance on.
In an even more surprising move, Glass gives Bridger a piece of advice that will change his life forever: "Follow your own lead, Bridger" (2.22.49). And that's just what Bridger does: the end of the novel sees him petitioning Captain Henry to be sent out on trapping mission deep in the wilderness, a mission that marks his transition into maturity. What's more, the mere fact that Bridger makes this request shows how much confidence he's gained already.
(Fun fact: the historical Jim Bridger is one of the most legit American explorers of all time. Guess Glass had the right idea.)