  Let $K= \mathbb{Q}(\sqrt{d})$ be a real quadratic field with $d$ having three
distinct prime factors. We show that the $2$-class group of each layer in the
$\mathbb{Z}_2$-extension of $K$ is $\mathbb{Z}/2\mathbb{Z}$ under certain
elementary assumptions on the prime factors of $d$. In particular, it validates
Greenberg's conjecture on the vanishing of the Iwasawa $\lambda$-invariant for
a new family of infinitely many real quadratic fields.
