Many energy, environmental, industrial, and microfluidic processes rely on the flow of polymer solutions through porous media. Unexpectedly, the macroscopic flow resistance often increases above a threshold flow rate in a porous medium, but not in bulk solution. The reason why has been a puzzle for over half a century. Here, by directly visualizing flow in a transparent 3D porous medium, we demonstrate that this anomalous increase is due to the onset of an elastic instability in which the flow exhibits strong spatiotemporal fluctuations reminiscent of inertial turbulence, despite the small Reynolds number. Our measurements enable us to quantitatively establish that the energy dissipated by pore-scale fluctuations generates the anomalous increase in the overall flow resistance. Because the macroscopic resistance is one of the most fundamental descriptors of fluid flow, our results both help deepen understanding of complex fluid flows and provide guidelines to inform a broad range of applications.