  Let $p$ be an odd prime, and fix integers $m$ and $n$ such that $0<m<n\leq
(p-1)(p-2)$. We give a $p$-local homotopy decomposition for the loop space of
the complex Stiefel manifold $W_{n,m}$. Similar decompositions are given for
the loop space of the real and symplectic Stiefel manifolds. As an application
of these decompositions, we compute upper bounds for the $p$-exponent of
$W_{n,m}$. Upper bounds for $p$-exponents in the stable range $2m<n$ and
$0<m\leq (p-1)(p-2)$ are computed as well.
