During the time of war also, they did not let Karna do war. They always insulted every given chance against Karna. Is it just because Karna is a Sut Putru? Or they just hate him that Karna was a strong warrior and supporter to Duryodhana. Or they believed caste system a lot? Or why?
Is this actually the case? For example, in book seven after Bhisma's death we have:
O Lord, brave Bhisma's fall drive all other thoughts from the mind of your sons and the sons of Pandu ... it was at that very moment like the sudden return of a stranger the Kurus recalled the only one who bore compare to Bhisma: Karna, best of all swordsman that have ever lived ... Karna, Karna, the chieftains cried ...
Karna is reckoned a bull in the herd of men worth twice any of his peers ... Passion had driven his words to Bhishma, "whilst you live, O son of Kuru, I will not fight. Either you wipe out the Pandavas in this great war and I take my leave of Duryodhana and depart for the forest or the Pandavas will kill you Bhisma and you will ascend to heaven. Then I will mount my chariot and by my own hand destroy every and any warrior you can name." So it was for ten days that Karna refused to fight.
This sounds that 'hatred' is the wrong word to apply and more like a deeply passionate quarrel on both sides.
Moreover, we know that Karna is the elder half brother of Yudhi.shthira and so in a sense, the war is two dharmas contesting each other; though, were Karna's parentage known the rules of dharma - ultimate or human - would determine the succession to the throne and avoid the war - but then we would not have the Mahabharata, and the affairs of men and gods are not so easily settled.