Aims. The shape of low-frequency radio continuum spectra of normal galaxies is not well understood, the key question being the role of physical processes such as thermal absorption in shaping them. In this work we take advantage of the LOFAR Multifrequency Snapshot Sky Survey (MSSS) to investigate such spectra for a large sample of nearby star-forming galaxies. Methods. Using the measured 150 MHz flux densities from the LOFAR MSSS survey and literature flux densities at various frequencies we have obtained integrated radio spectra for 106 galaxies characterised by different morphology and star formation rate. The spectra are explained through the use of a three-dimensional model of galaxy radio emission, and radiation transfer dependent on the galaxy viewing angle and absorption processes. Results. Our galaxies' spectra are generally flatter at lower compared to higher frequencies: the median spectral index αlow measured between ≈ 50 MHz and 1.5 GHz is -0.57 ± 0.01 while the high-frequency one αhigh, calculated between 1.3 GHz and 5 GHz, is -0.77 ± 0.03. As there is no tendency for the highly inclined galaxies to have more flattened low-frequency spectra, we argue that the observed flattening is not due to thermal absorption, contradicting the suggestion of Israel & Mahoney (1990, ApJ, 352, 30). According to our modelled radio maps for M 51-like galaxies, the free-free absorption effects can be seen only below 30 MHz and in the global spectra just below 20 MHz, while in the spectra of starburst galaxies, like M 82, the flattening due to absorption is instead visible up to higher frequencies of about 150 MHz. Starbursts are however scarce in the local Universe, in accordance with the weak spectral curvature seen in the galaxies of our sample. Locally, within galactic disks, the absorption effects are distinctly visible in M 51-like galaxies as spectral flattening around 100-200 MHz in the face-on objects, and as turnovers in the edge-on ones, while in M 82-like galaxies there are strong turnovers at frequencies above 700 MHz, regardless of viewing angle. Conclusions. Our modelling of galaxy spectra suggests that the weak spectral flattening observed in the nearby galaxies studied here results principally from synchrotron spectral curvature due to cosmic ray energy losses and propagation effects. We predict much stronger effects of thermal absorption in more distant galaxies with high star formation rates. Some influence exerted by the Milky Way's foreground on the spectra of all external galaxies is also expected at very low frequencies.