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 didn't cope properly if not at all, the first thing he does after the death of jason is putting jason's robin clothes in the batcave to remind himself of his failure ( similar to how he wears a bat costume because he used to be afraid of bats ), he then proceeded to go into crime fighting as usual releasing his pain ...
It was Jason's way of showing Batman that he was back and that he had forgiven him. A much smaller version of this symbol could also be seen on the wall (near the door) on the outside of the storage depot where Batman surrendered to Scarecrow.
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.
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.
But, rather than blaming The Dark Knight or Joker for his gruesome death, he blames himself. Infamously, during the "Death in the Family" story arc, Jason Todd discovers that the woman he had long believed to be his biological mother was actually a stepmother.
Jason has long resented Batman for refusing to kill the Joker, believing he chose the villain's life over his. However, the pair's relationship has somewhat repaired over time, as Red Hood even adopted his former mentor's no-kill rule while rejoining the Bat-Family.
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.
In reality though, the Joker has him locked up in an abandoned wing of Arkham Asylum and proceeds to further torture him for about a year - warping his mind and making him believe that Batman just simply didn't care about him, abandoned him, and quickly replaced him with Tim Drake (Todd doesn't know that Batman thinks ...
Jason harbors understandable resentment towards Bruce Wayne, but it's clear he still cares - apparent by a heartbreaking gift he just gave to Batman. Jason famously was murdered at the hands of the Joker in Batman: A Death in the Family by Jim Starlin, Jim Aparo, and Mike DeCarlo.
Learning that Batman had not avenged him by killing the Joker, Jason was enraged and vowed to kill the Joker himself, as well as do what he felt Batman should do but could not: clean up Gotham City in a far more brutal and lethal manner.
The classic story saw Batman fail to save Jason from the Joker, who beat Robin within an inch of his life with a crowbar, before finally killing him when he blew up the warehouse where Jason lay incapacitated inside.
In New Teen Titans #55 by Marv Wolfman and George Pérez, Nightwing found out about Jason's death through a mutual Titan, the perpetually arrogant and callous Danny Chase. Nightwing would later use Jason's death to justify firing Danny.
Jason Todd's death deeply affected Batman, pushing him to the brink and nearly causing his own demise. Batman's reckless behavior after Jason's death was fueled by guilt and self-blame, leading him to be vulnerable to danger.
Jason Todd's relationship with Batman turned sour as a result of his battles with criminals, which were almost suicidal. Batman decided to relieve Jason from his duties as Robin in order to allow him to mourn the death of his parents. Jason resented this and stormed out, refusing to discuss the issue of his parents.
The Joker beats Jason with a crowbar and restrains him and Haywood in the warehouse with a time bomb. Jason throws himself on the bomb to shield Haywood as the warehouse explodes. Batman arrives too late to save them, and Jason and Haywood die from their injuries.
In the DC Universe, few characters are as divisive as Red Hood. He's not a traditional hero, nor is he a villain. Instead, he occupies a gray area, using his skills and his willingness to break the rules to fight crime in his own way.
Jason was kidnapped and taken to an abandoned wing of Arkham Asylum, where he was tortured by the Joker, who manipulated him into hating Batman over the course of a year. Joker faked Jason's death by sending a tape to Batman in which Robin was apparently shot and killed.
So it's pretty much generally accepted that Jason was 15 when he died. In most depictions via comics or live action movies like Under the Red Hood or A Death in the Family, I can clearly see that Jason is a boy in his mid teens. Not many people debate this.
Jason Todd, the former Robin killed by the Joker in Batman: A Death in the Family, is revealed to have been resurrected by Ra's al Ghul via the Lazarus Pit. But the pit changes him and his emotions and he becomes the new Red Hood. His debut culminates in a fateful confrontation with those he feels have wronged him.
In the animated movie, it's clear why he doesn't want to kill the Joker. It's because he doesn't really care about him. He has no interest in personal revenge against him. He has plenty of opportunities to kill him at any point but doesn't.
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.
Batman has apologized and attempted to help Jason move past his trauma several times; even as recently as last year's 3 jokers miniseries, but Jason wouldn't meet him halfway.
