We present radio spectral analyses for a sample of 29 radio-quiet (RQ) and three radio-loud (RL) narrow-line Seyfert 1 galaxies (NLS1s) detected with the Australia Telescope Compact Array at both 5.5 GHz and 9.0 GHz. The sample is characterized by Lbol/LEdd > 0.15. The radio slopes in 25 of the 29 RQ NLS1s are steep (α5.5 − 9.0 < −0.5), as found in earlier studies of RQ high Lbol/LEdd AGN. This steep radio emission may be related to AGN-driven outflows, which are likely more prevalent in high Lbol/LEdd AGN. In two of the three RL NLS1s, the radio slopes are flat or inverted (α5.5 − 9.0 > −0.5), indicating a compact optically-thick source, likely a relativistic jet. Archival data at 3.0 GHz, 1.4 GHz, and 0.843 GHz are also compiled, yielding a sample of 17 NLS1s detected in three bands or more. In nine objects, the radio spectra flatten at lower frequencies, with median slopes of α5.5 − 9.0 = −1.21 ± 0.17, flattening to α3.0 − 5.5 = −0.97 ± 0.27, and to α1.4 − 3.0 = −0.63 ± 0.16. A parabolic fit suggests a median spectral turnover of ∼ 1 GHz, which implies synchrotron self-absorption in a source with a size of only a fraction of 1 pc, possibly a compact wind or a weak jet. Two objects show significant spectral steepening to α < −2 above 3 GHz or 5 GHz, which may suggest relic emission from past ejection of radio emitting plasma, of the order of a few years to a few decades ago. Finally, two objects present a single spectral slope consistent with star-forming activity.