Though he never voiced it, Batman did privately admit that he didn't go searching for
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.
Because Jason Todd is Batman's adopted son, and Batman failed to stop Joker from murdering Jason. He regretted not being able to save Jason.
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.
Is it Batman's fault that Jason died? Jason Todd's death is often seen as Batman's greatest failure, but DC claims Red Hood's subsequent turn to the dark side was never Bruce's fault.
Jason despised Batman for not avenging his death and replacing him with Tim Drake, so he takes his fury out on the new Robin, savagely beating his successor. Jason's jealousy gets the better of him when he assaults Tim in Teen Titans Vol 3 #29.
Despite all of their differences later in life, Jason didn't forget about fixing Bruce's father's watch. While he didn't feel comfortable giving the gift to Bruce face-to-face, the fact Jason gave it to him years later shows that he still cares and kept his promise from when he was a kid.
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!
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.
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.
Jarro grew attached to Batman and even often calls him “father”. He wanted to prove to Batman that he can be the best Robin and even convinced Batman to give him a chance. Jarro became the new Robin and was even said to be Batman's favorite Robin, admitted by Batman himself.
There's no doubt that recalling the fate of his parents is Bruce's biggest fear and regret. No matter who kills Batman's parents in different media, it's the catalyst to the person he becomes. Bruce spent all his childhood and early adulthood scared of remember what had happened.
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.
Dick's immediate reaction to Jason's death was like this: Shock, disbelief and anger for why he was not told. He was at Tamaran at the time Jason died, but that does not excuse Bruce, Alfred or anyone else not telling him immediately after knowing about his return.
This enraged Nightwing who proceeded to beat the Joker nearly to death. It was only thanks to Batman's intervention that the Joker survived, but the act scarred Nightwing.
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.
Origins. When the Joker became obsessing over destroying the Bat-Family, he decided to arrange for his face to be surgically removed. According to the Joker, he did this to symbolize how even without his, "Mask" he was still the joker where-as Batman's Robins and Bat-Girls' identities were only mask-deep.
The General Rule: Batman Does Not Kill
The question of whether Batman kills in the comics is a controversial one, as the character has been portrayed in different ways by various writers and artists over the years. However, the general consensus is that Batman does not kill, at least not intentionally or directly.
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.
Jason's anger was rooted in jealousy. He noted how "they", presumably referring to the superhero community, never seemed to approve of him as Robin but they welcomed and celebrated Tim.
Todd, during his time as the second Robin, perished at the hands of the Joker, only to be revived years later. He assumed the mantle of Red Hood and has been an uneasy ally of Batman ever since.
Although his relationship with Bruce never truly healed, it did mend after a while with Bruce forgiving Jason for his actions and allowing to join Batman Incorporated as Wingman.
Earth-51 Batman managed to kill the Joker to avenge Jason, but we all know what happens when you kill, the next one becomes significantly easier. Batman, finally realising that killing is the only permanent solution, proceeds to kill EVERY SINGLE VILLAIN ON EARTH-51.
Most sources put him at roughly 15 when he became Robin in pre-Flashpoint (so prior to the 2011 reboot) continuity.