Jesus bore our sins. He is the theological (or Biblical) context for understanding this phrase. The historical context is punishment for breaking Roman law. Those who hung beside Jesus were guilty. "Bearing your cross" is carrying the weight, or living out the consequences, of your own guilt.
Where sin is involved, we should not be surprised but, instead, should expect the following responses. It is only by God's grace and the knowledge of Jesus Christ that we run to Him when we sin.