Back at King's Landing, Tywin is writing something and also ignoring Tyrion. He's multitasking! Tyrion breaks the silence by passive-aggressively complimenting him on being Hand of the King instead of Tyrion. Tywin complains, "You brought a whore into my bed." He also has an objection about the way that Tyrion got authority and power, and then spent his time bedding harlots and drinking with thieves. Tyrion complains about Tywin not visiting him during his convalescence. Tywin doesn't care that much: "Maester Pycelle assured me your wounds were not fatal." Tyrion lists his many accomplishments of the last season, which include destroying an entire invading fleet and covering up for King Joffrey's cowardice. He ends with, "What do I want? A bit of bloody gratitude would be a start." Tywin is not impressed: "Jugglers and singers require applause. You are a Lannister." Tywin's not impressed with much, really. He's trying to run seven kingdoms (because it's not like Joffrey is capable of doing it), and three of them are currently at war with him.
Tyrion finally gets to the point: Jaime technically gave up his claim to Casterly Rock when he joined the Kingsguard, so Tyrion says that makes him the rightful heir. Tywin says they'll find him a better chamber than that tiny one he's in now, reward him somehow for Blackwater, and get him a better position. And he could get a suitable wife at some point. However, Tywin would like to make something clear: "I would let myself be consumed by maggots before mocking the family name and making you heir to Casterly Rock." Tyrion asks why because he's now in the position of giving other characters setup lines. So I'm just going to devote an entire paragraph to Tywin's response:
"Why? You ask that? You who killed your mother to come into the world? You are an ill-made spiteful little creature full of envy, lust and low cunning. Men's laws give you the right to bear my name and display my colors, since I cannot prove that you are not mine. And to teach me humility, the gods have condemned me to watch you waddle about wearing that proud lion that was my father's sigil and his father's before him. But neither gods nor men will ever compel me to let you turn Casterly Rock into your whorehouse. Go. Now. And speak no more of your 'rights' to Casterly Rock. Go."
I like that it's not just Tyrion's height that Tywin objects to here. It's mostly the whores. He emphasizes this by telling him, "The next whore I catch in your bed, I'll hang." So I guess he doesn't yet know about Shae, the latest whore that Tyrion has been shacking up with. Well, I say "latest" but she's been around for a while now. And they probably haven't done much shacking since Shae was installed as Sansa's handmaiden.