The baseline design of CLIC (Compact Linear Collider) uses X-band accelerating structures for its main linacs. In order to maintain beam stability in multibunch operation, long-range transverse wakefields must be suppressed by 2 orders of magnitude between successive bunches, which are separated in time by 0.5 ns. Such strong wakefield suppression is achieved by equipping every accelerating structure cell with four damping waveguides terminated with individual rf loads. A beam-based experiment to directly measure the effectiveness of this long-range transverse wakefield and benchmark simulations was made in the FACET test facility at SLAC using a prototype CLIC accelerating structure. The experiment showed good agreement with the simulations and a strong suppression of the wakefields with an unprecedented minimum resolution of 0.1 V/(pC mm m).