  We study 4-dimensional simply connected Lie groups $G$ with left-invariant
Riemannian metric $g$ admitting non-trivial conformal Killing 2-forms. We show
that either the real line defined by such a form is invariant under the group
action, or the metric is half conformally flat. In the first case, the problem
reduces to the study of invariant conformally K\"ahler structures, whereas in
the second case, the Lie algebra of $G$ belongs (up to homothety) to a finite
list of families of metric Lie algebras.
