  We investigate the time-resolved photoelectric response of WSe$_2$ crystals
on common glass and flexible polyimide substrates to determine the effect of
the dielectric environment on the speed of the photodetectors. We show that
varying the substrate material can alter the speed-limiting mechanism: while
the detectors on polyimide are RC limited, those on glass are limited by slower
excitonic diffusion processes. We attribute this to a shortening of the
depletion layer at the metal electrode/WSe2 interface caused by the higher
dielectric screening of glass compared to polyimide. The photodetectors on
glass show a tunable bandwidth which can be increased to 2.6 MHz with
increasing the electric field.
