We present a detailed study of the Faraday depth structure of four bright (>1 Jy), strongly polarized, unresolved radio-loud quasars. The Australia Telescope Compact Array (ATCA) was used to observe these sources with 2 GHz of instantaneous bandwidth from 1.1 to 3.1 GHz. This allowed us to spectrally resolve the polarization structure of spatially unresolved radio sources, and by fitting various Faraday rotation models to the data, we conclusively demonstrate that two of the sources cannot be described by a simple rotation measure (RM) component modified by depolarization from a foreground Faraday screen. Our results have important implications for using background extragalactic radio sources as probes of the Galactic and intergalactic magneto-ionic media as we show how RM estimations from narrow-bandwidth observations can give erroneous results in the presence of multiple interfering Faraday components. We postulate that the additional RM components arise from polarized structure in the compact inner regions of the radio source itself and not from polarized emission from galactic or intergalactic foreground regions. We further suggest that this may contribute significantly to any RM time variability seen in RM studies on these angular scales. Follow-up, high-sensitivity very long baseline interferometry (VLBI) observations of these sources will directly test our predictions.