Viscoelastic polymer solutions exhibit a variety of flow instabilities and in particular, in mixed shear and extensional flow, elastic turbulence. Coincident with the transition to turbulence is additional dissipation that, in porous flow, may be characterised as an increased apparent viscosity. We report elastic turbulence and apparent thickening in the flow of polymer solutions both in rock samples and in microfluidic analogues and we correlate the onset of thickening and turbulence with rheological measurements. Contrary to expectations, the characteristic relaxation time associated with the transition to turbulence is found to be independent of polymer concentration over the range studied (10c* ≲c≲ 100c*). Furthermore, this characteristic time scales with the square of molecular weight. Thus the characteristic time associated with the transition to turbulence is not the linear-viscoelastic timescale usually measured but rather scales as a dilute Rouse time despite being an entangled system.