  The Bogomolov Conjecture is a finiteness statement about algebraic points of
small height on a smooth complete curve defined over a global field. We verify
an effective form of the Bogomolov Conjecture for all curves of genus at most 4
defined over a function field of characteristic zero. We recover the known
result for genus 2 curves and in many cases improve upon the known bound for
genus 3 curves. For many curves of genus 4 with bad reduction, the conjecture
was previously unproved.
