Aims. We aim to constrain new starburst/AGN models of IRAS bright galaxies via their spectral energy distribution from the nearinfrared to the radio. To this end, we determine the radio spectra for a sample of 31 luminous and ultraluminous IRAS galaxies (LIRGs/ULIRGs). Methods. We present here new high frequency VLA observations at 22.5 GHz and 8.4 GHz and also derive fluxes at other radio frequencies from archival data. Together with radio data from the literature, we construct the radio spectrum for each source. In the selection of data we have made every effort to ensure that these fluxes neither include contributions from nearby objects, nor underestimate the flux due to high interferometer resolution. Results. From our sample of well-determined radio spectra we find that very few have a straight power-law slope. Although some sources show a flattening of the radio spectral slope at high frequencies, the average spectrum shows a steepening of the radio spectrum from 1.4 to 22.5 GHz. This is unexpected, because in sources with high rates of star formation, we expect flat spectrum, free-free emission to make a significant contribution to the radio flux at higher radio frequencies. Despite this trend, the radio spectral índices between 8.4 and 22.5 GHz are flatter for sources with higher values of the far-infrared (FIR)-radio flux density ratio, q, when this is calculated at 8.4 GHz. Therefore, sources that are deficient in radio emission relative to FIR emission (presumably younger sources) have a larger thermal component to their radio emission. However, we find no correlation between the radio spectral index between 1.4 and 4.8 GHz and q at 8.4 GHz. Because the low frequency spectral index is affected by free-free absorption, and this is a function of source size for a given mass of ionized gas, this is evidence that the ionized gas in ULIRGs shows a range of densities. Conclusions. The youngest LIRGs and ULIRGs are characterized by flatter average radio spectral indices from 1.4 to 22.5 GHz, and by a larger contribution to their high frequency, radio spectra from free-free emission. However, the youngest sources are not those that have the greatest free-free absorption at low radio frequencies. The sources in which the effects of free-free absorption are strongest are instead the most compact sources. Although these have the warmest FIR colours, they are not necessarily the youngest sources.