Its not simply that he's vicious...he's an idiot. He's making disastrous mistakes that are going to get them all killed.
Admittedly they cut out that part of the book where he's standing on the castle walls shooting starving refugees with a crossbow, killing them because they dared to beg their king for food. "What do they think I am, a baker?"
But Joffrey is insane. Medically insane. No one is arguing that. The drama comes from how everyone around him reacts to this. The courtiers who don't do anything. His own mother who refuses to stop him.
Its not even about whether Cersei can or cannot stop Joffrey (there comes a point where she admits she can't if she wanted to; now that he's king what little control she ever had over him is gone). Its that she doesn't even want to.
The thing that makes so many people hate Cersei throughout the book series is she is utterly unapologetic and remorseless for things Joffrey has done. Such as that beautiful scene in "The North Remembers" when Tyrion bluntly confronts her with it. Just the little lies she tells herself and he points out how absurd they are: "I tried to stop it!" "Did you? You failed." What part about "I tried" comes close to making up for the political catastrophe of killing Ned Stark? The Starks will be at their throats for a generation.
And basically, she explained it to Sansa later in the season. Cersei's view is that she *admits* Joffrey is a monster, but she has no choice but to defend him because he is her child.
...when the reality is *yes*, she always has the choice, and difficult as it is...yes, she should stop defending her own child for the sake of everyone else. Real-life families of serial-killers and mass murdering war criminals have to go through such decisions, but they still have those decisions and we still judge them by them.
The worst part, which is just utterly satisfying when you read it, is you realize that Cersei doesn't even know her own children. For all of her ranting about "I love my children" and "its her one redeeming quality"....she's living in a fantasy land.
Cersei spent 17 years obsessing and fantasizing over the great king Joffrey would grow up into. In ALL of that time, did she bother to make him take lessons in statecraft? He doesn't even know how to use a sword! Didn't she bother to hire instructors? Just what was she *doing* for 17 years? Twiddling her thumbs? This was no quick snap judgement; she raised Joffrey horribly, even if he was already medically a psychopath from his incest genes.
Meanwhile, you realize that *Cersei has no actual relationship with Myrcella and Tommen*. Oh she gloats over them, eats meals with them, but she isn't particularly close to them. But she *thinks* she is. The familial bond between Catelyn Stark and Arya Stark simply doesn't exist between Cersei and Myrcella. That's not to say they have a "bad" relationship, just a more distant one than Cersei wants to admit. Surprisingly common, actually; the rich out of touch aristocrat who let servants completely raise her child with zero input on her own, who gazes out the mansion window at the child and gloats about what a wonderful relationship they have.
Sometimes you may see me obsessing over Myrcella somewhat. That's because she's the utter vindication of how Joffrey is completely Cersei's fault and she *has* no redeeming qualities. Cersei *had* no relationship with her children, and Myrcella is old enough to start to realize this herself as the ultimate witness. All of this craziness that Joffrey causes....this is all Cersei's fault. Joffrey's just an animal. Cersei's the antagonist who let this whole war happen.
There's this point in book 2, after the Riot at King's Landing, when Bronn bluntly asks, "Joffrey just caused a city wide riot. He's shooting refugees with a crossbow in the streets. He *has* a younger brother, Tommen. Why *aren't* we trying to assassinate Joffrey ourselves to install Tommen? It would be far less effort on the balance, and Joffrey's dragging all of us down with him". -- and you realize how true what Bronn is saying is; he meant it as a matter-of-fact open question; why *don't* they kill an insane king who has a younger sane brother? The best guess I have is that Tyrion doesn't want to be a kinslayer, or that it would look bad for public propaganda. But I'm still not sure why Tyrion didn't kill him after the riot, other than that Cersei would never allow it.