Many times he went to face the Witch-king but time after time he was stopped whether it was his horse bolting off, Glorfindel warning him against it or his Steward advising him not to listen to the Witch-king's taunts which he in the end gives in to. My question is whether Earnur thought he could genuinely dispose of the Witch-king? An enemy he had been warned against fighting against time and time again.
His motivation in meeting the Witch-king was not due to thinking he could defeat him, but rather wounded pride. We see the first sign of this when Glorfindel warned him off:
These words many remembered; but Earnur was angry, desiring only to be avenged for his disgrace.
And we see a general description of his character:
Earnur was a man like his father in valour, but not in wisdom. He was a man of strong body and hot mood...
Then when the Witch-king issues his first challenge:
When Earnur received the crown in 2043 the King of Minas Morgul challenged him to single combat, taunting him that he had not dared to stand before him in battle in the North. For that time Mardil the Steward restrained the wrath of the king.
And finally at the Witch-king's second and last challenge:
Earnur had held the crown only seven years when the Lord of Morgul repeated his challenge, taunting the king that to the faint heart of his youth he had now added the weakness of age. Then Mardil could no longer restrain him, and he rode with a small escort of knights to the gate of Minas Morgul.
Whether he did or didn't think he could defeat the Witch-king is immaterial. Despite the warnings-off, despite everything, it was pride, anger and lack of wisdom that led him to Minas Morgul.
All references: Lord of the Rings Appendix A.
It wasn't so much that he thought he could kill the witch-king necessarily--more the unthinking/unreasoning pride and/or urge to go to battle with a foe who had taunted him. Just as the military prefers to recruit 18-year-olds who will rush into an inferno to save their comrades without regard to their own selves and will be eager to assault a dug-in enemy position in a frenzy of self-confidence, because they really don't appreciate deep down that this can kill them. Earnur knew he wanted to engage the enemy; how, and whether he really could pull it off, were secondary considerations.