Jess was afraid to show his drawings to his dad, because his dad did not approve of his hobby. As we learn in chapter 2:

He would like to show his drawings to his dad, but he didn't dare. When he was in first grade, he told his father than he wanted to be an artist when he grew up. He'd thought he would be pleased. He wasn't. 'What are they teaching in that damn school?' he had asked. 'Bunch of old ladies turning my son into some kind of a—' He had stopped on the word, but Jess had gotten the message. It was one you didn't forget, even after four years.

Jess wanted to be an artist--and he was clearly very talented--but his dad did not believe that this profession was masculine enough for his son. His father's disapproval deeply wounds Jess, so he is scared to make himself vulnerable (by showing his drawings) to someone who might respond by hurting his feelings even more. Jess is struggling to figure out what it means to be a man. He wants to prove his masculinity to his father--this explains why he tries so hard to win the race, a pursuit society considers masculine--but he also desperately wants to be an artist. Lost in his confusion, he protects himself by hiding his skill rather than displaying it to his dad.