We present broadband polarimetric observations of a sample of high-Faraday-rotation-measure (high-RM) active galactic nuclei (AGN) using the Karl. G. Jansky Very Large Array (JVLA) telescope from 1 to 2 GHz, and 4 to 12 GHz. The sample (14 sources) consists of very compact sources (linear resolution smaller than ≈5 kpc) that are unpolarized at 1.4 GHz in the NRAO VLA Sky Survey (NVSS). Total intensity data have been modeled using a combination of synchrotron components, revealing complex structure in their radio spectra. Depolarization modeling, through the so-called qu-fitting (the modeling of the fractional quantities of the Stokes Q and U parameters), has been performed on the polarized data using an equation that attempts to simplify the process of fitting many different depolarization models. These models can be divided into two major categories: external depolarization (ED) and internal depolarization (ID) models. Understanding which of the two mechanisms is the most representative would help the qualitative understanding of the AGN jet environment and whether it is embedded in a dense external magneto-ionic medium or if it is the jet-wind that causes the high RM and strong depolarization. This could help to probe the jet magnetic field geometry (e.g., helical or otherwise). This new high-sensitivity data shows a complicated behavior in the total intensity and polarization radio spectrum of individual sources. We observed the presence of several synchrotron components and Faraday components in their total intensity and polarized spectra. For the majority of our targets (12 sources), the depolarization seems to be caused by a turbulent magnetic field. Thus, our main selection criteria (lack of polarization at 1.4 GHz in the NVSS) result in a sample of sources with very large RMs and depolarization due to turbulent magnetic fields local to the source. These broadband JVLA data reveal the complexity of the polarization properties of this class of radio sources. We show how the new qu-fitting technique can be used to probe the magnetized radio source environment and to spectrally resolve the polarized components of unresolved radio sources.