  The Zarankiewicz number $z(b;s)$ is the maximum size of a subgraph of
$K_{b,b}$ which does not contain $K_{s,s}$ as a subgraph. The two-color
bipartite Ramsey number $b(s,t)$ is the smallest integer $b$ such that any
coloring of the edges of $K_{b,b}$ with two colors contains a $K_{s,s}$ in the
first color or a $K_{t,t}$ in the second color.
  In this work, we design and exploit a computational method for bounding and
computing Zarankiewicz numbers. Using it, we obtain several new values and
bounds on $z(b;s)$ for $3 \le s \le 6$. Our approach and new knowledge about
$z(b;s)$ permit us to improve some of the results on bipartite Ramsey numbers
obtained by Goddard, Henning and Oellermann in 2000. In particular, we compute
the smallest previously unknown bipartite Ramsey number, $b(2,5)=17$. Moreover,
we prove that up to isomorphism there exists a unique $2$-coloring which
witnesses the lower bound $16<b(2,5)$. We also find tight bounds on $b(2,2,3)$,
$17 \le b(2,2,3) \le 18$, which currently is the smallest open case for
multicolor bipartite Ramsey numbers.
