Streak cameras are widely used to record the spatio-temporal evolution of laser-induced plasma. A prototype of picosecond X-ray streak camera has been developed and tested by Commissariat à l'Énergie Atomique et aux Énergies Alternatives to answer the Laser MegaJoule specific needs. The dynamic range of this instrument is measured with picosecond X-ray pulses generated by the interaction of a laser beam and a copper target. The required value of 100 is reached only in the configurations combining the slowest sweeping speed and optimization of the streak tube electron throughput by an appropriate choice of high voltages applied to its electrodes.