Jess is very protective of his art, as it is basically an extention of himself, and Jess suffers from extremely low self-esteem. He is the only male child at home, he receives little to no support from his family, he is bullied at school, and based on all of this you have to consider that Jess would be pretty sensitive to having his work viewed and possibly criticized by others.
Jesse is extremely self-conscious about his drawing. He feels isolated from his peers and from his family for so many reasons. He doesn't "fit in" at school, coming from a poor family, being a quiet kid. He doesn't "fit in" at home because he is imaginative and artistic, not like the athletic , farm "son" he should be. He is terrified to show his drawing to anyone because he fears more ridicule.