In the final episodes of "Breaking Bad," Jesse had been collaborating with Hank and Gomez to "nail" Walt.
Though Hank and Gomez are now (after the shoot-out in the desert between Hank & Gomez and the Aryan Nation) dead and thus unable to corroborate any such claims Jesse might make in this regard, Hank's wife, Marie, could do so (she remained there while Jesse was living in their home as a "safe house").
In addition to Marie's direct testimony, there is also certainly a great deal of secondary and circumstantial evidence pointing to Jesse having "come clean" with Hank and even having risked his life to help Hank "nail" Walt. For example, a copy of Jesse's DVD confession - with Hank off-camera, posing questions to Jesse, etc. - might still exist. If, after being liberated by Walt, Jesse were to have run directly to the APD or DEA and turned himself in, he could have told a convincing and true story about how he had intended to burn down Walt's house, how Hank interrupted him, how Hank convinced him to become a police informant / collaborator, etc.
The scene where Hank and Gomez are murdered and Jesse is taken prisoner would also speak in Jesse's favor. His months of brutalization (evidenced by scars, etc.) while in captivity by the Aryan Nation, being forced to manufacture meth, would have likewise buttressed his claim to "victim" status and elicited the court's sympathy.
So why doesn't Jesse, upon being liberated by Walt at the beginning of "El Camino," immediately turn himself in to the authorities? At the very least, he could have claimed that Hank had promised him immunity (whether or not Hank had that authority is moot - the claim, alone, would have given Jesse the jury's sympathy). Forensic evidence (of the cage in which Jesse was housed for months) would have further supported Jesse's claim that he was an innocent party - and even a victim. In a "worst-case" scenario, he might have gotten a year of probation.
Additionally, some "insider" knowledge that Jesse possessed (the location of the buried housekeeper that Todd had murdered; the hidden money in Todd's apartment) would have further burnished Jesse's image.
And what, actually, were the charges that could have been levelled against Jesse? His only outstanding charge was throwing $500,000 out of his car window onto people's front lawns. As far as I recall, that was the only charge against him - and subsequent events (backed by hard, but also circumstantial, evidence) could place that in a very positive light for Jesse.