We report on compact and robust supercontinuum generation and post-compression using transmission of light through multiple thin solid plates at the SwissFEL X-ray free-electron laser facility. A single stage consisting of three thin plates followed by a chirped mirror compressor achieves compression of initially 30-fs pulses with 800-nm center wavelength to sub-10-fs duration. We also demonstrate a two-stage implementation to compress the pulses further to sub-5-fs duration. With the two-stage setup, the generated supercontinuum includes wavelengths ranging from 500 to 1100 nm. The multi-plate setup is compact, robust, and stable, which makes it ideal for applications at free-electron laser facilities such as pump-probe experiments and laser-arrival timing tools.