Collisionless shocks are ubiquitous in the Universe and often associated with a strong magnetic field. Here, we use large-scale particle-in-cell simulations of nonrelativistic perpendicular shocks in the high-Mach-number regime to study the amplification of the magnetic field within shocks. The magnetic field is amplified at the shock transition due to the ion-ion two-stream Weibel instability. The normalized magnetic field strength strongly correlates with the Alfvénic Mach number. Mock spacecraft measurements derived from particle-in-cell simulations are fully consistent with those taken in situ at Saturn's bow shock by the Cassini spacecraft.