We study the radio spectral properties of 2094 star-forming galaxies (SFGs) by combining our early science data from the MeerKAT International GHz Tiered Extragalactic Exploration (MIGHTEE) survey with VLA, GMRT radio data, and rich ancillary data in the COSMOS field. These SFGs are selected at VLA 3 GHz, and their flux densities from MeerKAT 1.3 GHz and GMRT 325 MHz imaging data are extracted using the 'superdeblending' technique. The median radio spectral index is $\alpha _{\rm 1.3\, GHz}^{\rm 3\, GHz}=-0.80\pm 0.01$ without significant variation across the rest-frame frequencies ~1.3-10 GHz, indicating radio spectra dominated by synchrotron radiation. On average, the radio spectrum at observer-frame 1.3-3 GHz slightly steepens with increasing stellar mass with a linear fitted slope of β = -0.08 ± 0.01, which could be explained by age-related synchrotron losses. Due to the sensitivity of GMRT 325 MHz data, we apply a further flux density cut at 3 GHz ($S_{\rm 3\, GHz}\ge 50\, \mu$Jy) and obtain a sample of 166 SFGs with measured flux densities at 325 MHz, 1.3 GHz, and 3 GHz. On average, the radio spectrum of SFGs flattens at low frequency with the median spectral indices of $\alpha ^{\rm 1.3\, GHz}_{\rm 325\, MHz}=-0.59^{+0.02}_{-0.03}$ and $\alpha ^{\rm 3.0\, GHz}_{\rm 1.3\, GHz}=-0.74^{+0.01}_{-0.02}$. At low frequency, our stacking analyses show that the radio spectrum also slightly steepens with increasing stellar mass. By comparing the far-infrared-radio correlations of SFGs based on different radio spectral indices, we find that adopting $\alpha _{\rm 1.3\, GHz}^{\rm 3\, GHz}$ for k-corrections will significantly underestimate the infrared-to-radio luminosity ratio (qIR) for >17 per cent of the SFGs with measured flux density at the three radio frequencies in our sample, because their radio spectra are significantly flatter at low frequency (0.33-1.3 GHz).