As Meher Manda already wrote for Bustle, the Mother of Dragons may be angry, but that doesn't mean Dany's going mad on Game of Thrones. To call her such is to equate "a woman’s rightful anger with 'crazy,'" Manda wrote. That's extremely correct, but this is also Game of Thrones, which means the writers may not have cared that turning Daenerys into her father the mad king is pretty out of character for Dany and a kind of sexist decision.
Since that does seem to be the direction the show is headed, though, let's take a look at just what drove Aerys II mad that could potentially reveal Dany's path as well. For one, years of ancestral relationships in the Targaryen family couldn't have been good for the stability of the blood line. Aerys married his own sister, which made Dany's mom also her aunt. It's widely believed that Targareyns are born with "the taint" of madness — although some have been driven to madness later in life.
That seems to have happened to Dany's father, Aerys II. Perhaps he was pre-disposed to it, but events in his life also seem to have triggered the madness. These events include being held prisoner during a short uprising and the loss of three of his sons. It's worth noting that if Dany's dragon Drogon dies, she will have lost her three dragon sons. If she follows in her father's footsteps, that could be a catalyst for madness.
Aerys is also believed to have been hearing voices in his head that made him increasingly paranoid. As a result, he would often target supposed "threats" and burn them alive. In the booklore, he had such a fascination with fire that burning people alive became a fetish for him. Aerys was eventually murdered by Jaime Lannister to protect King's Landing. Aerys had planned to burn the city down with wildfire, but Jaime stopped him in time. Aerys' final words, as seen in a vision by Bran, were "burn them all!"
It's also the same command Missandei gave Dany right before Cersei had Missandei executed. Dany's advisor said, "Dracarys," which is the order Dany gives her dragons when she wants them to release hellfire upon her opponents. Dany seems to think that the only way she's getting on the throne is if she destroys everything in her path — which, right now, includes all the innocent citizens of King's Landing. That's why many fans think she's following through on the Mad King's "burn them all" command. She is, in essence, becoming the Mad Queen, according to many fans and theories.
But there are also reasons to believe Dany will not become mad just because her father was. For one, the madness doesn't seem to have affected all Targaryens. Jon Snow seems OK. Maester Aemon was chill. Plus, Dany is a woman — and female Targaryens seem less affected by the madness that runs in their family.
Reddit user anycullen pointed out that one of the only Targaryen women to go mad was Haelaena Targaryen. She was driven to madness after being forced to pick which of her children was to be executed and then seeing it enacted in front of her. That would mess with anyone, Targaryen or not.
Others also believe that Aerys would have continued being a kind, good ruler if he didn't begin hearing voices. And, one fan theory is that the voices are Bran's fault. Fans saw how Bran warging into Hodor messed with his mind; perhaps Bran warged into a vision of Aerys and equally destroyed his mind. Perhaps Daenerys can escape the Targaryen madness not only because she's a woman, but because her father wasn't made mad by blood, he was made mad by Bran.
Just looking at Daenerys' father's history is not enough to determine her future. Plenty of people have terrible parents on this show and in real life and grow up to not be anything like them. Fans should give Dany the space to be mad about the death of her dragon, her soldiers, her best friend, without calling her "mad." After eight seasons on our screens, she deserves way better than that.