We study the C * -algebras associated with upper semi-continuous Fell bundles over second-countable Hausdorff groupoids. Based on ideas going back to the Packer-Raeburn 'stabilization trick', we construct from each such bundle a groupoid dynamical system whose associated Fell bundle is equivalent to the original bundle. The upshot is that the full and reduced C * -algebras of any saturated upper semi-continuous Fell bundle are stably isomorphic to the full and reduced crossed products of an associated dynamical system. We apply our results to describe the lattice of ideals of the C * -algebra of a continuous Fell bundle by applying Renault's results about the ideals of the C * -algebras of groupoid crossed products. In particular, we discuss simplicity of the Fell-bundle C * -algebra of a bundle over G in terms of an action, described by Ionescu and Williams, of G on the primitive-ideal space of the C * -algebra of the part of the bundle sitting over the unit space. We finish with some applications to twisted k-graph algebras, where the components of our results become more concrete.