How do Jem and Scout mature throughout the book? What dilemmas do they face that help their thinking better mature?

I think Jem most matures by watching the truth of the trial and how it affects his father. From the mob in front of the jail in chapter 15 to the trial's verdict in the early 20s, and his reaction to the trial, this young teen is getting a taste of the evil of real men. He is protective of his dad by this point because his dad has been through the criticism of friends and foes alike and has lost something it was obvious he should not have lost. This makes Jem aware of the world's inequities. Children have a notion that things should be fair. As many parents say... Life isn't fair.

Scout's greatest moment of maturity occurs in the last chapter when she learns through experience the lessons Atticus worked to teach throughout the book. She learned to walk in someone else's shoes. She examined from his porch what he must have been watching all along.