Bhishma actually spoke harsh words because he knew Karna was Kunti's son and did not want him to fight the war and also according to him, he spoke ill of the pandavas for no reason.
https://www.sacred-texts.com/hin/m06/m06124.htm
Thou art Kunti's son, not Radha's! Nor is Adhiratha thy father! O thou of mighty arms, I heard all this about thee from Narada as also from Krishna-Dwaipayana! Without doubt, all this is true! I tell thee truly, O son, that I bear thee no malice! It was only for abating thy energy that I used to say such harsh words to thee! O thou of excellent vows without any reason thou speakest ill of all the Pandavas!
Through pride, and owning also to thy companionship with the low, thy heart hateth even persons of merit! It is for this that I spoke such harsh words about thee in the Kuru camp! I know thy prowess in battle, which can with difficulty be borne on earth by foes! I know also thy regard for Brahmanas. thy courage, and thy great attachment to alms-giving! O thou that resemblest a very god, amongst men there is none like thee! For fear of intestine dissensions I always spoke harsh words about thee.
Bhishma even embraced Karna after causing the guards to be removed.
Hearing these words, the aged chief of the Kurus,the son of Ganga, whose eyes were covered with film slowly raising his eyelids, and causing the guards to be removed, and seeing the place deserted by all, embraced Karna with one arm, like a sire embracing his son, and said these words with great affection:--Come, come!
According to Drona, Karna boasts on every eve of battle but Karna retreats which is why he labelled Karna half a rathi.
Hearing this, Drona, that foremost of all wielders of weapons, said, 'It is even so as thou hast said. That is not untrue! He boasteth on the eve of every battle, but yet he is seen to retreat from every engagement. Kind (out of season) and blundering, it is for this that Karna, in my judgment, is only half a Ratha!'
https://www.sacred-texts.com/hin/m12/m12a002.htm
Karna also told Drona in his tutelage that he wanted Brahmastra so he could fight Arjuna.
Beholding that Dhananjaya was superior to every one in the science of weapons, Karna. one day approached Drona in private and said these words unto him, 'I desire to be acquainted with the Brahma weapon, with all its mantras and the power of withdrawing it, for I desire to fight Arjuna. Without doubt, the affection thou bearest to every one of thy pupils is equal to what thou bearest to thy own son. I pray that all the masters of the science of weapons may, through thy grace, regard me as one accomplished in weapons!' Thus addressed by him, Drona, from partiality for Phalguna, as also from his knowledge of the wickedness of Karna, said, 'None but a Brahmana, who has duly observed all vows, should be acquainted with the Brahma weapon, or a Kshatriya that has practised austere penances, and no other.'
