Linear birefringence measurements of dilute and semi-dilute polyisobutylene solutions following flow through a disordered fixed fibre bed of 2.47% solids volume fraction provide both transient and steady measurements of chain deformation. Our results indicate that the flexible polyisobutylene polymers undergo a large conformation change, stretching in the direction of the average flow. This occurs even though the average flow in the bed is a plug flow which would not cause any polymer stretch by itself. The polymer stretch or conformation change increases with the number of chain interactions with bed fibres and ultimately reaches a steady-state value that can be correlated with the pore-size Deborah number (i.e. a characteristic polymer relaxation time divided by a characteristic flow time in the bed pore). Large changes in the polymer conformation are noted for values of the Deborah number, De > 5. In addition, the time to steady state scales with the characteristic flow time within a pore over a large range of Deborah numbers. The pressure drop across the fibre bed was also measured simultaneously with the birefringence measurement and was found to be directly proportional to the birefringence throughout the range of De investigated. Thus, we show empirically, for the first time, that chain elongation, which produces normal stress anisotropy within the fluid, is directly responsible for the increased flow resistance. These findings are then analysed in the light of recent theories for the response of polymer molecules in fixed bed flow fields (Shaqfeh & Koch 1992). It is shown that our results are consistent with the interpretation that these flows are stochastic strong flows, which create an apparent ‘coil-stretch’ transition. After extending the theory of Shaqfeh & Koch to account for the specifics in the experiments, including the bed geometry and statistics as well as the polydispersity of the polymer solutions, it is shown that the theory can predict most of the experimental results both qualitatively and quantitatively.