  We show that the distance of a link $K$ with respect to a bridge surface of
any genus determines a lower bound on the genus of essential surfaces and
Heegaard surfaces in the manifolds that result from non-trivial Dehn surgeries
on the knot. In particular, knots with high bridge distance do not admit
non-trivial non-hyperbolic surgeries or non-trivial cosmetic surgeries. We
further show that if a knot has bridge distance at least 3 then its bridge
number is bounded above by a function of Seifert genus, or indeed by the genus
of (almost) any essential surface or Heegaard surface in the surgered manifold.
