One of the most interesting features in the relationship between the Lannister siblings is how they are each confirming and strengthening the suspicions they have about each other. This dynamic becomes notable for the first time in „A Clash of Kings“, when we can see from Tyrion’s POV how his actions are misperceived by Cersei, but how Tyrion at the same turn is absolutely blind to how his actions are perceived (while we as readers remain salient about this fact). Nowhere is this more evident as in Tyrion’s scheme to poison Cersei into suffering from diarrhea, where Cersei (as will later be confirmed in her POV) thinks he’s out to destroy her family and her.
Tyrion, of course, has his own suspicions about Cersei, is mistrustful and prepares. Matters come to a head when Tyrion brings Tommen in his possession because he doesn’t trust Cersei’s security measures and mistakenly thinks it gives him an added layer of protection. Cersei retaliates with full force (moderation not being her thing and all), bringing Alalaya in her possession, which only leads to Tyrion ushering threats, which of course Cersei takes as a sweeping confirmation of all his guilt (and likely to the order to Mandon Moore to kill Tyrion).
Tyrion likely only thought about hurting Cersei on a personal level, but the volonqar-prophecy hanging over her head leads her to assume that he’s on an omnicidial mission to destroy all of her and her children in one fell swoop. It’s only his impotence in „A Storm of Swords“ that prevents the escalation ladder to move forward on Tyrion’s part, but Cersei only suspects worse plots – which are of course reinforced by every uninhibited utterance from Tyrion, by every attempt of the imp to retaliate against Joffrey. Tyrion doesn’t understand why Cersei protects her vile son against all sense, and how can he? The volonqar doesn’t have a clue about the prophecy he takes center stage in.
But the toxic relationship also swirls up Jaime when he returns to King’s Landing. After his rash confession to Tyrion after the rescue, Tyrion hurls poisonous insults at Cersei (and Jaime), which lead Jaime to weigh every word and action of Cersei’s in „A Feast for Crows“ on Tyrion’s scale, seeing confirmation of Cersei’s adultery in every action (and rightly so), getting further and further estranged from his sister until he finally throws her plea for help into the fires of a winterly hearth.
Even Cercei’s governance in „A Feast for Crows“ is dominated by seeing Tyrion’s hand in everything that happens, any problem that comes up. How can he not be the driving factor, given that his hands are about to choke her perfect white neck? Of course, she finds proof for these assumptions all the time, which lead her to make Lord Bronn into an enemy, lose allies left and right and ultimately lose her position totally.
But it would be too much to attribute this wholly to the three Lannister children alone. It was the loving nurturing of Tywin, who perceived the whole existence of Tyrion as a slight, no matter what his son did, and to assume his daughter was only a glorified brood mare no matter what she did. No wonder the kids turned out as damaged as they are, confirming each other’s suspicions as they go.