We report the direct quantification of molar mass degradation in the drag-reducing polymers polyethylene oxide (PEO) and polyacrylamide (PAM) in turbulent pipe flows with an upstream tapered contraction. We find that entrance effects associated with the upstream contraction dominate the polymer degradation. Quantifying degradation according to the scaling relationship γ̇w∝ Mws-n the exponent n is determined to be -2.20±0.21 and -2.73±0.18 for PEO and PAM, respectively. Here Mws is the steady-state (or limiting) weight-average scission molar mass. A methodology is devised to circumvent polymer degradation due to the upstream contraction and thereby conduct degradation experiments in which only the turbulent flow in the pipe is responsible for chain scission. In this case, the scission-scaling relationship for PEO is γ̇w∝ Mw-3.20±0.28. Here Mw is the degraded weight-average molar mass after one pass through the 1.63-m length of pipe. Based on these scaling relationships we obtain a new upper limit for polymer drag reduction that is determined by chain scission rather than the maximum drag reduction asymptote.