Barth on Knowledge on Sin
It is true that all of Barth's theology stems from reflection upon Jesus Christ. Attached to this is our absence of knowledge as creatures apart from revelation by our Creator. For Barth this applies to all things—including our own inherent sinfulness.
Barth mingles with the law/gospel relationship, precisely because knowledge of God must proceed knowledge of our own sinfulness,
"What I have just said about man's incompetence is not an anthropological judgment drawn from the knowledge of man in himself, it is not a pessimistic conclusion imposed by the examination of the human situation taken in itself. It is the quite simple consequence of the knowledge that God has given himself to us, it is also the expression of our acknowledgment of this. He who has known the gift of God cannot do otherwise than say: 'God has found me, though I have been incapable of finding him. He has loved me, although I have been a rebel. By granting me the resurrection from the dead, he has shown me that I was dead.' Such a confession as a criterion of the truth, of the presence of the Holy." (126-127)
Thinking about knowledge of sin this way reveals that even the law's conviction of sin is grace. It is predicated on the knowledge that God has given Himself for us to redeem us. For Barth, there is not an abstract general statement of man's sinfulness.
So while these thoughts might disrupt a traditional Lutheran understanding of Law/Gospel, so also it speaks against legalistic tendencies in Reformed preaching. Barth rejects "knowledge of man in himself" and thus the attempt to turn a man's eyes inward on his own sinfulness. We do not prove man's sinfulness by looking at the disruptions of man's own actions in themselves. For in themselves, there is no judgment of the actions. The judgment of those actions as sin is decreed by God's revelation—particularly for Barth in the giving of Jesus Christ. None of this stands opposed to the Reformed Tradition.
Yet sometimes, "original sin" or "total depravity" in the Reformed Tradition is espoused in a way that (apparently if not truly) considers man apart from God. They present a message that requests listeners to perform, as Barth rejects, an "examination of the human situation taken in itself." This is sometimes unintentionally presented as conviction from some neutral "law" or examples of worldly sinfulness. Unintentionally, these implicitly ignore that knowledge of sin is only ever a gift of revelation from God. Yes, we know mankind is sinful, but we do so because of God's gracious revelation to us.
Understanding this, Barth the preacher rarely ever presents "law" before "gospel." For it is only in the grace of the gospel that one can respond to the former reality found in the law.