  Shimura curves on Shimura surfaces have been a candidate for counterexamples
to the bounded negativity conjecture. We prove that they do not serve this
purpose: there are only finitely many whose self-intersection number lies below
a given bound.
  Previously, this result has been shown in [BHK+13] for compact Hilbert
modular surfaces using the Bogomolov-Miyaoka-Yau inequality. Our approach uses
equidistribution and works uniformly for all Shimura surfaces.
