  We present Karl G. Jansky Very Large Array (VLA) S- (2--4 GHz), C- (4--8
GHz), and X-band (8--12 GHz) continuum observations toward seven radio-loud
quasars at $z>5$. This sample has previously been found to exhibit spectral
peaks at observed-frame frequencies above $\sim$1 GHz. We also present upgraded
Giant Metrewave Radio Telescope (uGMRT) band-2 (200 MHz), band-3 (400 MHz), and
band-4 (650 MHz) radio continuum observations toward eight radio-loud quasars
at $z>5$, selected from our previous GMRT survey, in order to sample their
low-frequency synchrotron emission. Combined with archival radio continuum
observations, all ten targets show evidence for spectral turnover. The turnover
frequencies are $\sim$1--50 GHz in the rest frame, making these targets
gigahertz-peaked-spectrum (GPS) or high-frequency-peaker (HFP) candidates. For
the nine well-constrained targets with observations on both sides of the
spectral turnover, we fit the entire radio spectrum with absorption models
associated with synchrotron self-absorption and free-free absorption (FFA). Our
results show that FFA in an external inhomogeneous medium can accurately
describe the observed spectra for all nine targets, which may indicate an FFA
origin for the radio spectral turnover in our sample. As for the complex
spectrum of J114657.79+403708.6 at $z=5.00$ with two spectral peaks, it may be
caused by multiple components (i.e., core-jet) and FFA by the high-density
medium in the nuclear region. However, we cannot rule out the spectral turnover
origin of variability. Based on our radio spectral modeling, we calculate the
radio loudness $R_{2500\rm\, \AA}$ for our sample, which ranges from
12$^{+1}_{-1}$ to 674$^{+61}_{-51}$.
