I have heard that toothbrushes should be disinfected with chlorine bleach or running them through the dishwasher. Is that a good method to keep from having to replace toothbrushes so often?

No, putting the toothbrush in hot water in the dishwasher or under the faucet, will reduce the life of the brush. Bleaching the toothbrush leaves a chlorine residue on the bristles which affects the taste the next time the brush is used.