Jem is slightly embarrassed by his younger sister, especially the pugilistic tendencies she demonstrates in the playground. As there is a four year age gap between the children it is natural that there is a distance between them. Jem is a little more aware of the social propriety that their father Atticus would like them to show, so he is more restrained in his behaviour, and he leaves the endless questioning of the injustices of their society to Scout.
There is a strong bond between the children despite their distance at school. Their endless summer games are together, with their curious neighbour, Dill. When Scout's life is in danger during the cowardly attack from Bob Ewell, Jem unhesitatingly comes to her aid. They are very much typical siblings.
Jem and Scout have the typical older brother-younger sister relationship. Jem is a little more serious about things than Scout is. Scout is quick tempered and quick to speak whatever is on her mind. Jem on the other hand is more reserved. Jem doesn't want to have anything to do with Scout at school because he sees her as the aggravating little sister. Jem thinks of himself as the big man at the school because he is four years older than Scout.
Jem is a little embarrassed by the way Scout reacts to people. Scout will fight anyone at anytime. Jem doesn't think she should act this way. We can see that Jem has a lot of Atticus in him. Jem and Scout are certainly close, though. During the summers, the two of them, plus Dill, have the best times together. They look out for each other. They are each other's confidants. Jem and Scout have no idea just how strong their bond is. They can't even imagine what is waiting for them, and that they are going to have to rely on each other to make it through.
I love how Harper Lee made the relationship of Jem and Scout so real. She gets us invested in their lives and we want only good things for them. We can relate to how Jem feels, having a younger sibling who drives us crazy, yet we would do anything in the world for them.