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We present a Faraday rotation spectroscopy (FRS) technique for measurements on the micron scale. Spectral acquisition speeds of many orders of magnitude faster than state-of-the-art modulation spectroscopy setups are demonstrated. The experimental method is based on charge-coupled-device detection, avoiding speed-limiting components, such as polarization modulators with lock-in amplifiers. At the same time, FRS spectra are obtained with a sensitivity of 20 $\mu$rad (0.001$^\circ$) over a broad spectral range (525 nm - 800 nm), which is on par with state-of-the-art polarization-modulation techniques. The new measurement technique also automatically cancels unwanted Faraday rotation backgrounds. Using the setup, we perform Faraday rotation spectroscopy of excitons in a hBN-encapsulated atomically thin semiconductor WS$_2$ under magnetic fields of up to 1.4 T at room temperature and liquid helium temperature. We determine the A exciton g-factor of -4.4 $\pm$ 0.3 at room temperature, and -4.2 $\pm$ 0.2 at liquid helium temperature. In addition, we perform FRS and hysteresis loop measurements on a 20 nm thick film of an amorphous magnetic Tb$_{0.2}$Fe$_{0.8}$ alloy., Comment: 8 pages, 4 figures