How does the relationship between Scout and Jem change over the course of the novel?

Jem and Scout start off as play buddies and pretty much equals when it comes to fighting each other. With Jem being four years older than she is, he hits puberty first and understands the world around them a little bit better. As Jem makes this natural transformation from childhood to adolescence, there is a shift to an unequal relationship. For example, as the trial of Tom Robinson nears in chapter 14, Jem recognizes the strain it is placing on everyone in the house. In an effort to help solve this problem, he says to Scout the following:

"Scout, try not to antagonize Aunty, here? . . . he's (Atticus) got a lot on his mind now, without us worrying him. . . it's this Tom Robinson case that's worryin' him to death--" (137-138).

Scout doesn't see this talk as anything but Jem trying to boss her around, though, and she challenges him to the point that he threatens to spank her if she won't stop. This is the first time that he's threatened to say and do something that seems so above her--as if he's the adult and she's the child. The next thing he does, though, completely kicks him out of the realm of childhood. When Jem tells Atticus that Dill has run away from his home in Meridian to their house, Scout explains what happened next:

"Dill's eyes flickered at Jem, and Jem looked at the floor. Then he rose and broke the remaining code of our childhood. He went out of the room and down the hall. 'Atticus,' his voice was distant, 'can you come here a minute, sir?'" (141).

This is a turning point for Jem as he moves into the adult world of understanding how to deal with crises. He even tells Dill that he shouldn't worry his mother the way he did by running away. And as far as Scout's relationship with him, it seems to have greatly changed as a result. Scout doesn't hold Jem's transformation against him for too long, though. By chapter 28, Scout says the following:

"Jem was becoming almost as good as Atticus at making you feel right when things went wrong" (259).

Jem winds up being a good big brother who does his best to defend Scout and keep her safe. Their relationship changes as all do, but they certainly go through a few growing pains along the way.