Jess keeps his drawing supplies hidden because he is a little embarrassed by them, and Jess is definitely afraid of what his dad would think. Readers get a little insight into Jess's relationship with his father in chapter 1. We see that Jess craves his father's attention and approval, but Jess hasn't been getting it lately. The narrator tells readers that Jess desperately wants to be the fastest kid around and that it would win him his father's approval:

Maybe Dad would be so proud he'd forget all about how tired he was from the long drive back and forth to Washington and the digging and hauling all day. He would get right down on the floor and wrestle, the way they used to. Old Dad would be surprised at how strong he'd gotten in the last couple of years.

It makes readers think about why Jess's father wouldn't currently be proud of his son. Chapter 2 provides readers with the answer. We are told that Jess loves to draw. He has a passion for it and a gift for it as well. We are also told that Jess would love to show his dad some of his drawings, but Jess doesn't dare do that anymore. When Jess was in first grade, he told his dad that he wanted to be an artist when he grew up. Jess's father was absolutely not supportive of such a dream and career. In fact, Jess's father was insulting about the notion entirely. The response stung Jess, and he has been afraid to open up that part of himself to his father ever again. He hides his materials because he doesn't want to further disappoint his father:

He would like to show his drawings to his dad, but he didn't dare. When he was in first grade, he had told his dad that he wanted to be an artist when he grew up. He'd thought his dad would be pleased. He wasn't. "What are they teaching in that damn school?" he had asked. "Bunch of old ladies turning my only 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.

Why does Jess hide his drawing supplies under his mattress in Bridge to Terabithia?

Jess keeps his drawing supplies under his mattress because he knows that his father doesn't approve of his hobby. He also doesn't dare show his father his drawing supplies because he does not want his father to be angry with him.

When he was in the first grade, he remembers telling his father that he wanted to be an artist when he grew up. His father had gotten very angry upon hearing his son's words. Jess remembers how painful the exchange had been; his father had basically implied that the teachers at school were turning him into a wimp.

The irony of it is that none of Jess' regular teachers actually support his artistic endeavors. In fact, they 'would screech about wasted time, wasted paper, wasted ability' when they caught him drawing. The only one who supports him is his music teacher, Miss Edmunds.