Single-phase flows of viscoelastic polymer solutions in both microfluidic devices and rock cores exhibit apparent flow thickening. We demonstrate that this thickening occurs above a critical Deborah number corresponding to the onset of spatio-temporal fluctuations. These fluctuations are observed to occur over a broad range of spatial and temporal scales consistent with elastic turbulence. The fluctuations provide a previously unreported mechanism for enhancing the displacement of a second, capillary trapped, immiscible phase.