  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.
