We establish a relation between the "large r" asymptotics of the Turaev-Viro invariants T V-r and the Gromov norm of 3-manifolds. We show that for any orientable, compact 3-manifold M, with (possibly empty) toroidal boundary, log vertical bar T V-r(M)vertical bar is bounded above by Cr parallel to M parallel to for some universal constant C: We obtain topological criteria for the growth to be exponential; that is log vertical bar T V-r(M)vertical bar >= Br, for some B > 0, and construct infinite families of hyperbolic 3-manifolds whose Turaev-Viro invariants grow exponentially. These constructions are essential for related work of the authors which makes progress on a conjecture of Andersen, Masbaum and Ueno. We also show that, like the Gromov norm, the values of the invariants T V-r do not increase under Dehn filling. Finally we give constructions of 3-manifolds, both with zero and non-zero Gromov norm, for which the Turaev-Viro invariants determine the Gromov norm.