Narrow-line Seyfert 1 (NLS1) galaxies are active galactic nuclei (AGN) believed to be in the early stages of their evolution. A fraction of them have been found to host relativistic jets. Due to the lack of large-scale diffuse radio emission they are believed to be experiencing one of their first activity cycles, and can offer us an opportunity to study the early evolution of more powerful AGN, such as radio galaxies and flat-spectrum radio quasars. Recently, a group of intriguing jetted NLS1s was discovered: based on high radio frequency data they host relativistic jets, but in the JVLA observations they all showed steep radio spectra at least up to 9.0 GHz, indicating very strong absorption at these frequencies. In this paper we study a subset of these sources in detail by employing spatially resolved radio spectral index maps at central frequencies of 1.6, 5.2, and 9.0 GHz. With spectral index maps we can disentangle the different radio emission components over the radio-emitting region, and get insights into the production mechanisms of radio emission. In addition, we study their host galaxies in relation to the radio emission to investigate if the host can provide us additional information regarding the origin of the radio emission, or the launching mechanism of the jets. It is fascinating how different the sources studied are, and certainly more, especially wide frequency-range, and high-resolution observations will be needed to understand their history and current properties, such as the reason behind the extraordinary radio spectra.