  By combining high-sensitivity LOFAR 150MHz, uGMRT 400MHz and 1,250MHz, GMRT
610MHz, and VLA 5GHz data in the ELAIS-N1 field, we study the radio spectral
properties of radio-detected star-forming galaxies (SFGs) at observer-frame
frequencies of 150-5,000MHz. We select ~3,500 SFGs that have both LOFAR 150MHz
and GMRT 610MHz detections, and obtain a median two-point spectral index of
$\alpha_{150}^{610}=-0.51\pm0.01$. The photometric redshift of these SFGs spans
$z=0.01-6.21$. We also measure the two-point radio spectral indices at
150-400-610-1,250MHz and 150-610-5,000MHz respectively for the GMRT
610-MHz-detected SFGs, and find that, on average, the radio spectrum of SFGs is
flatter at low frequency than at high frequency. At observer-frame
150-5,000MHz, we find that the radio spectrum slightly steepens with increasing
stellar mass. However, we only find that the radio spectrum flattens with
increasing optical depth at $V$-band at $\nu<1$GHz. We suggest that spectral
ageing due to the energy loss of CR electrons and thermal free-free absorption
could be among the possible main physical mechanisms that drive the above two
correlations respectively. In addition, both of these mechanisms could
physically explain why the radio spectrum is flatter at low frequency than at
high frequency.
