To me the respective arcs of Jaime and Tyrion are possibly the most astounding stuff about the series. If you think about it, they’ve nearly completely switched their “alignment” for lack of a better term.
In AGoT, the beginning, Tyrion is somewhat lovable, sort of a scamp character, who doesn’t bear anyone any ill will, he gets along with Jon after a rough start, designs a custom saddle for Bran, and generally is not what most people would consider a bad person, despite the many lumps in life he’s taken. Certainly he’s cynical, and devious, but I don’t recall him doing anything bad to anyone just because he’s a bad person.
By the end of ASoS, he’s just taken too many slaps to the face from the events of the books, with perhaps the final blow being Jaime revealing to him that his first wife, who he was convinced was a paid whore acting out a part, was actually what she seemed to be, an innocent girl who loved him. This coming from possibly the only person who he genuinely loves (Jaime), is the last straw, and he tells Jaime a terrible lie (that he did kill Joffrey) just to hurt him, and then kills his father and the whore who he loved and protected up until he found her in his father’s bed. On top of all this, he’s reviled in all of the Seven Kingdoms as a kingslayer and kinslayer, unjustly.
So he’s taken a huge journey from cynical but not evil to outright “fuck the world”.
Contrast this with Jaime, who starts AGoT as completely without empathy, living basically only for the thrill of combat and fucking his sister. Very nearly the first thing we see him do is throw Bran out the window. He’s arrogant, self-centered, and cares nothing for honor. He has some sense of duty or loyalty to his Lannister family and does seem to bear some affection for his children that he cannot acknowledge.
At the end of ASoS, he’s humbled by having been crippled and has completely changed his priorities and outlook on life. The first thing he does in King’s Landing is refuse to resume his relationship with Cersei, and shortly afterwards he refuses his father’s request/demand that he set aside his white cloak and oaths. He continues this new “honorable” Jaime trend by doing things like trying to do a proper job as Lord Commander of the Kingsguard, setting Brienne off on her journey with a sword named Oathkeeper that is forged from Ned Stark’s sword, and freeing his brother from prison.
His journey is from vile feckless youth to tragic, battered, but resolute man who seems determined to live a new life of honor.
