  The bipartite Ramsey number $b(s,t)$ is the smallest integer $n$ such that
every blue-red edge coloring of $K_{n,n}$ contains either a blue $K_{s,s}$ or a
red $K_{t,t}$. In the bipartite $K_{2,2}$-free process, we begin with an empty
graph on vertex set $X\cup Y$, $|X|=|Y|=n$. At each step, a random edge from
$X\times Y$ is added under the restriction that no $K_{2,2}$ is formed. This
step is repeated until no more edges can be added. In this note, we analyze
this process and show that the resulting graph witnesses that $b(2,t) =\Omega
\left(t^{3/2}/\log t \right)$, thereby improving the best known lower bound.
