How is it that Tywin can bitch-slap his own children so much (forced incest, forced marriage, screwing over Tyrion constantly) but has never said a stern word to Joffrey? I have a hard time believing he could actually be afraid of him considering he's bankrolling his armies. Maybe he's trying to keep him on his good side so he can be impressionable? It would be unsurprising that this causes the Lannisters to be the only family completely wiped out after the war.
And how does he expect any babies to be made with big mama Tyrell's grandkids considering the match-ups.