how did david discovered the true steerforth in the hardest ways

 The one thing David dislikes about Steerforth is the way he treats Mr. Mell, who has been very kind to David.  One day, when Mr. Creakle is gone and the boys are being particularly rowdy, Mr. Mell tells them to sit down but Steerforth defies him. Steerforth then makes fun of him for having his mother in a poor house, which was revealed to him by David, who had been told when Mr. Mell picked him up. Mr. Creakle comes in and, unsurprisingly, sides with Steerforth, resulting in Mr. Mell's departure. David feels extremely guilty about the whole exchange but gets over it quickly, trusting in Steerforth's judgment and not wanting to get on his bad side like Traddles, who stands up for Mr. Mell and earns Steerforth's disfavor in return.