The death of Jason Todd was one of the worst things to ever happen to Batman and the Bat-Family at large, changing how Batman views his Robins, and unfortunately, led to the death of another Robin, Stephanie Brown.
After the Joker killed Jason Todd, Batman practically sealed himself off from his extended family. He refused to take another Robin, stopped interacting with Oracle and Nightwing, and barely even spoke to Alfred. Bruce devoted himself entirely to his crusade against crime, believing that he should only risk his own life against criminals. Unfortunately, this also led to Batman getting significantly darker in how he approached criminals. Batman's new ultra-violent approach eventually drew the attention of a young Tim Drake, who came to believe that Batman needed a Robin to prevent him from going off the deep end. Batman did open up to the idea of more Robins, but the trauma of Jason's death would lead him to turn his back on Stephanie Brown, the often-forgotten fourth Robin, which directly led to her death.
Batman's Firing Of Stephanie Brown Sealed Her Fate
After Tim Drake stepped away from the role of Robin, Stephanie Brown took up the mantle, making her the only canon female Robin. Batman trained her for several months before letting her out in the field, but ultimately due to her performance, Batman fired Stephanie Brown from the role of Robin. This was devastating to Stephanie and in order to prove her worth to Batman, she stole some of his plans from the Batcomputer and tried to enact them herself. If the plans had succeeded, it would have brought all crime in Gotham under the direct control of Batman; the only problem was the plan needed Batman's alternate persona of Matches Malone to manipulate the gangs. Stephanie wasn't aware of who Matches really was, which led to an all-out gang war across Gotham City.
Stephanie had been fighting crime for years at this point, she started out as Spoiler and eventually became Robin. There were numerous times when both Tim and Batman told her to give up crime fighting, and she never did. Batman was afraid Stephanie would be killed fighting crime, so he fired her from being Robin. During her time as Robin, she showed stubborn behavior and ignored Batman's direct orders for what she felt was the greater good. She also showed a willingness to use lethal force, both traits Jason Todd had. It's likely that Batman saw Jason's behavior in Stephanie and fired her to prevent her from being killed; Batman even explicitly states after firing her that he expects her to give up crime fighting.
Stephanie Brown Died Because Batman Pushed Her Away
Batman wasn't able to see that fighting crime was something Stephanie was never going to stop, just like Jason would never stop being a criminal unless someone intervened. Batman stopped Jason from going down a dark path by letting him be Robin, but with Stephanie, Batman turned his back on her. He justified this by saying that if she continued being Robin, she'd die as Jason Todd did, but because Batman rejected her, it only pushed her further down the path of dying to crime.
Ultimately, Stephanie Brown did everything she could as both Robin and later Batgirl to fight crime, and she did it without the support of anyone, not Batman and not Tim, which is ultimately what got her killed just like Jason Todd.