Guys are ferociously independent. Their sense of 'guyness' is wrapped up in being able to take care of things. For many guys, to admit they can't handle a situation hits at the core of who they are. They may want to run away rather than admit that they can't do it.
This is not about not trusting the girl. Most guys have a fierce protective streak, they are afraid that if they admit their inadequacy the girl will lose faith in their abilities as a 'protector' - maybe even seek out a stronger, more capable person. Fundamentally, the woman's right to choose her man makes guys afraid to admit their flaws. This is not helped by the girl saying "I accept you with all your flaws" because the guy knows that is a lie - a girl will always assess her guy for flaws first, before even beginning the relationship. A guy never thinks of his girl giving him unconditional love and support because that doesn't happen. A girl's affection is conditional on your doing 1001 things right... if you do 2 wrong she may ditch you without much trouble.
Telling him you want to share his worries will NOT help... it takes far more trust and security in a relationship to share trouble. Like maybe 25 years of marriage and four kids.
BTW: this is normal for ALL guys. The rare ones who shared their insecurities with women during the caveman era got pushed out of the evolutionary race by women who felt "These guys are not good enough protectors". Women have a responsibility, in a sense, to ensure the quality control of the species. For most of pre-history (and history) that translated to 'ensure the guy is a good protector'. It'll be a while before evolution overcomes this.