I've read in many places that he was hesitant to punish his killers, which eventually led to the fitnah and then his own death. If he'd had punished them possibly the battle of Karbala could've been averted. So can anyone please tell?
It must be considered that Uthman was not killed by a single person, whom can be sentenced to death by the Islamic ruler. Massive revolution by the Muslims, ultimately, led to death of Uthman. Therefore, Imam Ali (PBHU) could not punish tens of thousands of revolutionary people. A revolution that Uthman himself was not blameless in creating it.
In confirmation of this, the book "Siffin war" states that:
" Abu Amayma bahli (ابو امامه باهلى) and Abu Darda (ابو درداء ), who were in Levant, came to Muawiya, and said to him: "O Muawiya!, on what basis are you fighting with Ali (AS)? We swear to Allah that he was Muslim before you became Muslim, he is more worthy than you in ruling Muslim's society, and he is closer to the Prophet Muhammad (PBHU) than you. So why do you fight with him?
Muawiya responded: I fight against him in revenge of Uthman's blood, because Ali has sheltered Uthman's murderers. Tell him to permit us to reach Uthman's murderers and show them to us. In which case I will be the first person dwelled in Levant with whom I swear allegiance."
Then, they went to Ali (AS) and informed him of Muawiya's statement. Ali (AS) said: "If you want Uthman murderers, this is what you see." There were twenty thousand people or more who were wearing iron armor, except that their eyes could not be seen and they were saying: We are all his killers, they should take retaliation for all of us if they with to retaliate. So Abu Amaymah and Abu Darda returned and did not attend any of the wars, and this happened in the month of Rajab. "
- This question can also be raised if Muawiya really wanted to defend the oppressed, why did not he rise up to defend and take revenge of Hazrat Hamzah (AS)? Is there anything else but he and his father interfered in that? And was not Uthman's death only an excuse for Muawiya to pursuit his political intentions?
 المنقرى، نصر بن مزاحم، وقعة صفین، ص 191، تحقیق، عبد السلام محمد هارون، القاهرة، المؤسسة العربیة الحدیثة، الطبعة الثانیة، افست قم، منشورات مکتبة المرعشى النجفى، 1404 هـ. ق.
In the name of Allah, the Entirely Merciful, the Especially Merciful
Briefly speaking, De facto, Othman was not murdered by just an individual, that he/she can be able to be punished by Islamic ruler, but there was a great revolution from Muslims that ultimately was led in murdering of him. As a result, Imam Ali (a.s.) couldn’t punish tens thousands of people that Othman himself was not without fault in that.
Muawiye and the appointed governor of Egypt were not really concerned about the murderers of Othman. Their main objective was getting the caliphate for themselves. The umma chose Ali (rdlhn) as the khalifa to succeed the martyred Othman.
This is verified by the fact that Muawiye never followed up on this investigation of who killed Othman soon after the death of Ali. Muawiye was the caliph after the death of Ali and he signed a non-succession agreement with Ali's oldest son Hasan.
Othman's relatives asked Muawiye at that point to bring the murderers of Othman to justice as late justice is better than no justice and Muawiye had rebelled against Ali for Ali's reluctance in this regard. Muawiye then changed his mind and never sought the justice for Othman's family as he was never sincere in this in the first place.
Ali had appointed his own sons, Hasan and Hussein to defend Othman but they were overwhelmed by the mob that entered Othman's home. Ali was more interested in preserving the unity of the umma than adding fuel to the flames of rebellion. If Muawiye had not caused the fitna, things would have calmed down and this justice would have been pursued in accordance with Sharia. Muawiye and his cohorts never allowed the Islamic state to form and instead caused fitna and civil war.