Strange as the title may seem, it is true that it was Bhimasena who vowed to kill Karna, through Arjuna of course. Similarly, it was Bhimasena who vowed that Shakuni will be killed by Sahadeva.
Popular legends have it that Bhima, during the game of dice, vowed to break the thigh of Duryodhana and drink the blood of Dushashana. It is also wrongly assumed that Arjuna at the same time vowed to kill Karna and Sahadeva vowed to kill Shakuni. The TV serials also show events in this way. But what actually occured was completely different.
It was Sri Madhvacharya, in his magnum opus Mahabharata Tatparya Nirnaya, who highlighted the fact that it was Bhima who made all the three vows just after the game of dice. Even the original Mahabharata has the same story. Unfortunately, Mahabharata has been diluted and distorted significantly and this very important aspect has been twisted.
During the game of dice, Bhimasena is the only who protests against the crime (from the Pandavas side of course – along with Vidura from the Kauravas). In fact, he gets extremely upset over Yudhishthira and asks Sahadeva to bring some fire so that he can “burn the hands of Yudhishtira”. When Arjuna objects, Bhima tells him that when elders commit mistakes, insulting them by words is equal to punishing them. The same principle is mentioned by Lord Krishna later during the 17th day of the war when Arjuna intends to kill Yudhishthira (yes, Arjuna tries to kill Yudhisthira! – more about this in another article)!
Later, when the Pandavas lose the game of dice a second time and start to move out of the palace, Dushashana and Duryodhana, aided by Karna and Shakuni, start to heap insults on the Pandavas. At that time, Bhima once again reiterates his vow concerning Dushashana and Duryodhana. And then, he makes two very interesting vows. I quote from the translation available at sacred texts.
“I will slay Duryodhana, and Dhananjaya will slay Karna, and Sahadeva will slay Sakuni that gambler with dice. I also repeat in this assembly these proud words which the gods will assuredly make good, if ever we engage in battle with the Kurus, I will slay this wretched Duryodhana in battle with my mace, and prostrating him on the ground I will place my foot on his head. And as regards this (other) wicked person–Dussasana who is audacious in speech, I will drink his blood like a lion”
It is only after this that both Arjuna and Sahadeva take their vows. They both acknowledge Bhima and tell the audience that, as directed by Bhima, they shall slay Karna and Sahadeva.
In the Mahabharata, truly, while Lord Krishna is the “director”, Bhimasena is the “screenplay writer” who ensures that the events unfold as per Krishna’s plans.