  A manifold with an irreducible $SO(3)$-structure is a $5$-manifold $M$ whose
structure group can be reduced to the group $SO(3)$, non-standardly imbedded in
$SO(5)$. The study of such manifolds has been initiated by M. Bobie\'nski and
P. Nurowski who, in particular, have shown that one can define four
$CR$-structures on a twistor-like $7$-dimensional space associated to $M$. In
the present paper it is observed that these $CR$-structures are induced by
almost contact metric structures. The purpose of the paper is to study the
problem of normality of these structures. The main result gives necessary and
sufficient condition for normality in geometric terms of the base manifold $M$.
Examples illustrating this result are presented at the end of the paper.
