What Jung prescribed is more specific than the popular version, and the difference matters.
The instruction is not to defeat the shadow, and not to express it. It is to recognise it, to withdraw the projection by which its contents are seen in other people and admit that they belong to oneself. Jung's clinical observation was that unacknowledged shadow material reliably reappears as an intense, disproportionate reaction to some quality in someone else, and he treated that reaction as diagnostic. He also held that the shadow contains genuine vitality and that a person wholly separated from it is diminished rather than purified.
Fleeing the figure in a dream, on his reading, is the situation continuing. Turning to face it is the work beginning. This is where his account of the shadow and the common chase dream connect, and it is why the two entries are often collapsed.
Several cautions are in order.
The concept is not empirically established. It is a clinical and interpretive framework, generated from case material and from Jung's reading of myth, alchemy and comparative religion. It has been enormously influential in psychotherapy, in literary criticism and in popular self-understanding, and it has not been demonstrated in the way an experimental finding is demonstrated. It does not follow that it is false. It follows that it should be labelled.
The popular usage has also drifted. Shadow work as now marketed is a large commercial category with only a loose relation to Jung's writings, and much of what is sold under the name would not have been recognised by him. Where a modern account describes the shadow as a set of traumas to be healed, it has substituted a different psychology for his while keeping his vocabulary.