Infrared-faint radio sources (IFRS) are a class of radio-loud (RL) active galactic nuclei (AGN) at high redshifts (z > 1.7) that are characterised by their relative infrared faintness, resulting in enormous radio-to-infrared flux density ratios of up to several thousand. We aim to test the hypothesis that IFRS are young AGN, particularly GHz peaked-spectrum (GPS) and compact steep-spectrum (CSS) sources that have a low frequency turnover. We use the rich radio data set available for the Australia Telescope Large Area Survey fields, covering the frequency range between 150 MHz and 34 GHz with up to 19 wavebands from different telescopes, and build radio spectral energy distributions (SEDs) for 34 IFRS. We then study the radio properties of this class of object with respect to turnover, spectral index, and behaviour towards higher frequencies. We also present the highest-frequency radio observations of an IFRS, observed with the Plateau de Bure Interferometer at 105 GHz, and model the multi-wavelength and radio-far-infrared SED of this source. We find IFRS usually follow single power laws down to observed frequencies of around 150 MHz. Mostly, the radio SEDs are steep, but we also find ultra-steep SEDs. In particular, IFRS show statistically significantly steeper radio SEDs than the broader RL AGN population. Our analysis reveals that the fractions of GPS and CSS sources in the population of IFRS are consistent with the fractions in the broader RL AGN population. We find that at least 18% of IFRS contain young AGN, although the fraction might be significantly higher as suggested by the steep SEDs and the compact morphology of IFRS. The detailed multi-wavelength SED modelling of one IFRS shows that it is different from ordinary AGN, although it is consistent with a composite starburst-AGN model with a star formation rate of 170 solar masses per year.
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