  We calculate the bridge distance for $m$-bridge knots/links in the $3$-sphere
with sufficiently complicated $2m$-plat projections. In particular we show that
if the underlying braid of the plat has $n - 1$ rows of twists and all its
exponents have absolute value greater than or equal to three then the distance
of the bridge sphere is exactly $\lceil n/(2(m - 2)) \rceil$, where $\lceil x
\rceil$ is the smallest integer greater than or equal to $x$. As a corollary,
we conclude that if such a diagram has more than $4m(m-2)$ rows then the bridge
sphere defining the plat projection is the unique minimal bridge sphere for the
knot.
