We demonstrate the operation of a compact x-ray free-electron laser (FEL) using the optical klystron mechanism. This effect speeds up the FEL process using the dispersion of magnetic chicanes placed between the undulator modules of the FEL beamline. The demonstration was performed at the soft x-ray beamline of SwissFEL, called Athos, which is, as far as we are aware, the only x-ray FEL to date with magnetic chicanes between every two undulator modules. In our measurements, we show that, compared to standard operation without chicanes, the required undulator length to achieve FEL saturation is reduced between 15% and 30% for radiation wavelengths between 1 and 2 nm. Fully exploiting the optical klystron effect represents an important step toward more compact FEL designs, rendering this key technology more affordable and hence accessible to a larger science community.