  We consider an infinite family of real quadratic fields $k$ where the
discriminant has three distinct odd prime factors, and the prime 2 splits. We
show that the unramified Iwasawa module $X(k_{\infty})$ associated with the
$\mathbb{Z}_2$-extension of $k$ has a bounded quotient. Thus, we also verify
Greenberg's conjecture on the vanishing of Iwasawa invariants for such fields
and obtain a finer structure for $X(k_{\infty})$.
