  We prove that any smooth projective geometrically connected non-isotrivial
curve of genus $g\ge 2$ over a one-dimensional function field of any
characteristic has at most $16g^2+32g+124$ torsion points for any Abel--Jacobi
embedding of the curve into its Jacobian. The proof uses Zhang's admissible
pairing on curves, the arithmetic Hodge index theorem over function fields, and
the metrized graph analogue of Elkies' lower bound for the Green function. More
generally, we prove an explicit Bogomolov-type result bounding the number of
geometric points of small N\'eron-Tate height on the curve embedded into its
Jacobian.
