  Meier and Zupan introduced bridge trisections of surface links in $S^4$ as a
4-dimensional analogue to bridge decompositions of classical links, which gives
a numerical invariant of surface links called the bridge number. We prove that
there exist infinitely many surface knots with bridge number $n$ for any
integer $n \geq 4$. To prove it, we use colorings of surface links by keis and
give lower bounds for the bridge number of surface links.
