The Joker kept
Though he never voiced it, Batman did privately admit that he didn't go searching for Jason because he was afraid of what he would find. He knew Jason was killing people, and Batman couldn't live with the shame of knowing that he had unintentionally created a deadlier version of Joe Chill.
He does so not as Jason Todd but as Red Hood… ironically the name the Joker once went by. Batman explains to him that he has a “no-kill” code of ethics and therefore he couldn't avenge his death.
How did Joker torture Jason Todd? Joker faked Jason's death & sent it to Batman, ensuring that Batman would never try and find Jason. Then, he kept him in Arkham Asylum and tortured him for almost 2 years.
In the story, Batman: Three Jokers, Jason was able to get his revenge sort of. He shot the Joker, who is the Clown, in the head.
After capturing Scarecrow and returning to Panessa Studios, the main floor between the cells was painted with a large, red logo of Red Hood. It was Jason's way of showing Batman that he was back and that he had forgiven him.
edit: now i believe he was 15 or 16 years old, no more and no less, that's the age of the trigger for puberty and at that age he builds the muscle mass he had in the movie. Edit2: Did you guys realize how cruel the Joker is? He tortured an underage teenager to death!
TLDR: Superman didn't save Jason because he was being tortured by Lex Luthor at the time in order for Luthor to create Conner Kent/Superboy. This is why he wants nothing to do with Superboy once he appears.
Fans were quick to despise this more vengeful version of Robin. Although Jason was motivated by good intentions, he displayed a ruthlessness that was uncharacteristic of the Boy Wonder.
They also interrogated Joker's henchmen who were with him during Todd's murder, in hopes to find out how the boy could have survived. Talia later restored Todd's health and memory by immersing him in a Lazarus Pit in which her father was also bathing and helped him escape the House of al Ghul.
Jason would remain the second Robin until he was kidnapped and beaten to a pulp by The Joker, who left him in a warehouse to die via explosion, Bruce attempted to save him, but he was too late, and Jason died, leaving Bruce to mourn the lost of his son.
On Earth- 51, Batman went on to kill the Joker after Jason Todd's death and eventually killed off a majority of his world's Supervillains in ways that looked like accidents.
Jason remained dead for six months, his legacy haunting Batman. However, when the evil-alternate reality Superman known as Superboy-Prime punched reality itself, the timeline stuttered, and Jason was restored to life.
Killing the Joker would only make the people of Gotham think that killing is always an option for stopping criminality. Batman doesn't want people to have that kind of mindset. As such, he knows that killing the Joker would only lead to more criminals rising up to become criminals as well.
A Death in the Bat-Family
“A Death in the Family,” which ran in Batman #426-429, hinges on a chilling scene where the Joker ambushes Robin (Jason Todd) and beats the Boy Wonder senseless with a crowbar, leaving him tied up in a warehouse with a bomb.
The Joker Does Not Care Who Is Behind The Mask
None of this matters to The Joker, however, as he's far more concerned with costumed personas than the ordinary people underneath them, so to him, killing Jason Todd was as much an attack on his and Batman's lighthearted era as killing Dick Grayson would be.
He suffers from bipolar disorder and chronic, suicidal depression, and wrestles constantly his history of abuse and how it affects him today. All of his mental instability, too, was only compounded by the eldritch power that rests within the Lazarus Pit that brought about resurrection.
The main thing that Jason Todd's Robin was criticized for is his impulsive nature and tendency to jump headfirst into a dangerous situation, something Batman repeatedly chastised him for. This aspect of Jason's personality has spilled over into his life as Red Hood.
Fandom. What if Jason Todd had never died? Batman wouldn't have become as dark and depressed after and Tim probably wouldn't have tried to become Robin immediately, although he probably would have figured out Bruce's identity. Jason, however, did have an anger problem which possibly led to the death of Felipe Garzonas.
Batman's “no killing” rule is to ensure he always acts in service of justice, not vengeance, and that he never let himself become so above and beyond the law that the last restraint on his actions is removed.
It's discovered that the Joker has a chemical curse that comes with his death. An airborne virus that infects whoever is nearest to him when he dies.
Batman is convinced that if he kills the Joker he'll sink to his level and become something like him. Joker echoes this sentiment frequently while badgering Batman about their many similarities. If Batman really believed he was as limitless as he says he is, he wouldn't be so scared of losing himself so easily.
The poll received 10,614 votes and 5,343 voted for Jason's death over 5,271 for his survival—a margin of just 72 votes. Although Kahn dispelled rumors that the process was rigged in favor of Jason's demise, O'Neil said it was possible many votes favoring Jason's death came from a single person.
After six months of torture, the Joker finally ends his video series by shooting Jason Todd in the chest, leading Batman to believe that his protégé had been killed. However, it is later revealed that the Joker had made Jason Todd wear body armor during the shooting, and that he had indeed survived.
After Tim went missing for three weeks, Batman discovered him at the abandoned Arkham Asylum, where it turned out The Joker tortured him for information and brainwashed him into being a mini-Joker.