HBO's Game of Thrones prides itself on complex characterization. Audiences can go from being made to fall in love with a pure character doing heroic deeds to hating them in the next episode, where they commit an atrocity for an unjustified or unknown reason.
Jaime Lannister goes through the opposite arc, where he starts out as a character that fans despise and morphs into one that many adore. Despite his shift from complete monster to an understandable man who's a product of the society he grew up, there are still many things it's hard to understand about the Kingslayer.
Why He Doesn't Tell People The Truth
When he's only seventeen, Jaime Lannister murders the king he's sworn to protect He has a good reason, as far as regicide justification goes — the king is planning to commit genocide, destroying a city of half a million people by burning them alive. But when it's discovered what Jaime has done, he doesn't bother to tell anyone about this plan, instead letting them think him a monster.
It's one heck of a defense mechanism, and he should swallow his pride and tell people the truth. Less would hate him.
Why He Pushes Bran
Jaime has his own justification for pushing Bran Stark from a window — he'll tell Robert Baratheon about the affair that has produced bastards and passed them off as heirs to the Iron Throne. But even though fans can kind of understand why Jaime jumps to such a conclusion, it's hard to imagine why he finds it so easy to push a child from a window, or why he almost seems to take pride and pleasure in doing it.
Why He Insults Brienne
Jaime and Brienne's relationship is the ultimate enemies-to-lovers trope of the show, but why he starts out insulting her is... beyond fans. It's probably his defense mechanisms, of course, but even so, he makes fun of everything from her appearance to her sword-fighting skills. It's mean and highly unnecessary.
Why He Hates Ned
Ned Stark despises Jaime, because he values honor and Jaime killing a king doesn't fit his definition of honor — or many people's, to be fair.
It's another situation where the tension might have been eased if Jaime was willing to swallow his pride and tell Ned exactly what happened with Aerys Targaryen, but he'd rather hate the man instead.
Jaime's defense mechanisms are the root of all his problems.
Why He Doesn't Stand Up To Tywin Sooner
Tywin Lannister is a man who hates his youngest son, blaming him for the death of Joanna, his mother. It takes a long time for Jaime to be willing to stand up to Tywin in regards to this treatment, to the point where he waits until Tyrion is on the brink of death but — why? Why wait this long? Jaime doesn't have much of a problem defying Tywin when it comes to marrying and producing an heir, so...
Why He Isn't Attached To His Kids
Jaime has three children by Cersei and although he's around them every day, he seems to find it easy to distance himself (at least at the beginning of the series). It's hard to imagine why he isn't more attached to them, even unwillingly. They are half-him and half-the woman he loves, yet he wants little to do with them.
It's a bit understandable when it comes to Joffrey, but Tommen and Myrcella are always very sweet.
Why He Doesn't Kill Robert Baratheon
It's not that he should kill Robert Baratheon but a question of why he doesn't.
Robert is abusive towards Cersei (though Cersei tells Ned she hides this from Jaime, who'd kill him). Still, how hasn't Jaime found out and murdered the man who abuses his sister and children? Jaime has no qualms with killing kings, he's proven that in the past.
Why He's So Enamored With Cersei
It's truly hard to imagine why he's so enamored with Cersei. Is it vanity, because she's his twin, or something else? Cersei is beautiful but also spiteful and manipulative and, having been around her for his entire life, it's hard to believe that Jaime doesn't know this.
Even so, he sticks by her side. It's admirable... in a way.
Why He Goes Back To Her
He finally does end up breaking away from Cersei, coming to realize she's an awful person — finally!
But in the end, he leaves Brienne — a woman so much better for him, and not related to him, as an added bonus — and returns to his sister. Considering the series builds him up for an identity arc where he realizes he doesn't need Cersei, it's an insult to his character to have him return. Yet he does.
Why He Ends Up This Way
It's hard to puzzle Jaime out. It's difficult to understand why he ends up having something of a good heart beneath that violent exterior, but also how someone with such a good heart can push children out of windows and commit awful deeds. Jaime, in general, is just a complicated character who doesn't always make a whole lot of sense.