In To Kill a Mockingbird, why did Jem have such a difficult time accepting how the trial ended?

The primary reason Jem struggles to accept the guilty verdict given to Tom is because he is unwilling to accept the fact that in his society, justice does not always win out.
Throughout the novel Lee builds events in the story in a way in which Scout and Jem (and therefore the read) have to learn tough lessons about kindness and mercy and justice. What's more, these lessons were learned for characters who are considered "lower" in society. Scout learns to be kind and merciful to her schoolmate who is poor. Scout and Jem learn to respect and defend Boo Radley. Atticus even sends the message of mercy when he puts down the rabid dog. The author instills this message throughout the novel and builds an expectation of justice and fairness.
When Tom receives a guilty verdict, despite sound evidence and a moving speech from Atticus, Jem is not able to reconcile what he knows and expects about justice with the reality of justice in his time. Jem also experiences a pang of resentment towards his father, who has expected kindness and mercy and justice from his own children, but was unable to execute it himself.

Jem is idealistic and naïve to the prejudices and unfairness of the world and how these biases will impact the jurors. It's his idealism that causes him to feel overly confident and optimistic that his father will produce an acquittal for Tom Robinson based on the testimony heard throughout the trial.
As Reverend Sykes predicted, Tom Robinson is found guilty and Jem witnesses for the first time the reality of the injustices of the judicial system when it pertains to colored people. The strong beliefs he once held in the goodness of people is now false. The realization that there is true evil within their society shakes Jem to the core and he is unable to accept the jury’s unfair conviction. He must now reevaluate his understanding of human nature and come to terms with the disappointing realities of inequality and racism. As Atticus once told Jem that the courtroom was a place where everyone is seen as equal, Jem now knows that his father was wrong.

Throughout the trial, Jem carefully listens as his father presents moving, logical arguments proving Tom Robinson's innocence. However, Jem is a naive child and does not take into consideration the significant impact that the jurors' prejudice will have on the verdict. Jem takes note of the Ewells' conflicting testimonies, witnesses his father prove that Bob Ewell was motivated to assault his daughter, considers the fact that Tom Robinson is handicapped, and understands the importance of Bob being left-handed, which corresponds to the location of Mayella's injuries. In Jem's mind, Atticus clearly proves Tom's innocence and he firmly believes that Tom will be acquitted. Before Judge Taylor reads the verdict, Jem even tells Reverend Sykes in confidence that Atticus will win the case.
Unfortunately, Tom is wrongly convicted of assaulting and raping Mayella, which destroys Jem's childhood innocence and upsets him. Before Judge Taylor read the verdict, Jem believed that he lived in a just, fair society. After witnessing profound racial injustice for the first time, Jem's naive belief in the justice system and Maycomb's "Christian" ideals is destroyed. He immediately begins to cry and continually tells Atticus that the verdict was not fair. Jem ends up becoming a jaded individual, who can no longer trust his fellow citizens after witnessing Tom's wrongful conviction.