Well, there are several factors leading to her "bitchiness"
- First of all, she's Lord Tywins oldest child (only a few minutes older than Jaime, but still).
However, due to being a woman, she was never regarded as anything more than a possible marriage alliance. On several occasions she states how she longed to be a man (not sexually, just socially), for as a woman, all sorts of
direct rule are closed to her.
While that's debatable (ask the Mormonts or the Lady of Thorns

), it's still been nagging on her for a looong time. It's the most likely source of her unhealthy hunger for power. She's completely desperate to prove that a woman can be just as good & strong a ruler as any man.
--> One factor leading to her hate for the Tyrells, because their alliance with the throne would reduce her to a mere mother & lady of court again
--> Also leading to her overprotectiveness, because she realizes that her only chance to rule is through her children (so she wants to rule out all outside influence).
Unrelated:
The second big factor, is a prophecy *possible "Feast" spoilers*, made by a maegi in her youth, basically telling her that 1. another woman, more young and beautiful and 2. the "Valonqar" (someone close to her) will be her downfall.
She expects Maergary to be the first one
--> second factor leading to her hate for the Tyrells
and Tyrion (or at this point maybe even Jaime) to be the Valonquar
--> explaining her behaviour towards/fear of Tyrion, especially after his Outbreak and maybe to some extent her behaviour towards the "new" Jaime. And of course her overall paranoia