We report a high-speed (~2 kHz) dynamic multiplexed fiber Bragg grating (FBG) sensor interrogation using a wavelength-swept laser (WSL) with a polygon-scanner-based wavelength filter. The scanning frequency of the WSL is 18 kHz, and the 10 dB scanning bandwidth is more than 90 nm around a center wavelength of 1,540 nm. The output from the WSL is coupled into the multiplexed FBG array, which consists of five FBGs. The reflected Bragg wavelengths of the FBGs are 1,532.02 nm, 1,537.84 nm, 1,543.48 nm, 1,547.98 nm, and 1,553.06 nm, respectively. A dynamic periodic strain ranging from 500 Hz to 2 kHz is applied to one of the multiplexed FBGs, which is fixed on the stage of the piezoelectric transducer stack. Good dynamic performance of the FBGs and recording of their fast Fourier transform spectra have been successfully achieved with a measuring speed of 18 kHz. The signal-to-noise ratio and the bandwidth over the whole frequency span are determined to be more than 30 dB and around 10 Hz, respectively. We successfully obtained a real-time measurement of the abrupt change of the periodic strain. The dynamic FBG sensor interrogation system can be read out with a WSL for high-speed and high-sensitivity real-time measurement.