  We aim to understand the physical mechanisms that drive star formation in a
sample of mass-complete (>10$^{9.5}M_{\odot}$) star-forming galaxies (SFGs) at
1.2 $\leq z$ < 1.6. We selected SFGs from the COSMOS2020 catalog and applied a
$uv$-domain stacking analysis to their archival Atacama Large
Millimeter/submillimeter Array (ALMA) data. Our stacking analysis provides
precise measurements of the mean molecular gas mass and size of SFGs. We also
applied an image-domain stacking analysis on their \textit{HST} $i$-band and
UltraVISTA $J$- and $K_{\rm s}$-band images. Correcting these rest-frame
optical sizes using the $R_{\rm half-stellar-light}$-to-$R_{\rm
half-stellar-mass}$ conversion at rest 5,000 angstrom, we obtain the stellar
mass size of MS galaxies. Across the MS (-0.2 < $\Delta$MS < 0.2), the mean
molecular gas fraction of SFGs increases by a factor of $\sim$1.4, while their
mean molecular gas depletion time decreases by a factor of $\sim$1.8. The
scatter of the MS could thus be caused by variations in both the star formation
efficiency and molecular gas fraction of SFGs. The majority of the SFGs lying
on the MS have $R_{\rm FIR}$ $\approx$ $R_{\rm stellar}$. Their central regions
are subject to large dust attenuation. Starbursts (SBs, $\Delta$MS>0.7) have a
mean molecular gas fraction $\sim$2.1 times larger and mean molecular gas
depletion time $\sim$3.3 times shorter than MS galaxies. Additionally, they
have more compact star-forming regions ($\sim$2.5~kpc for MS galaxies vs.
$\sim$1.4~kpc for SBs) and systematically disturbed rest-frame optical
morphologies, which is consistent with their association with major-mergers.
SBs and MS galaxies follow the same relation between their molecular gas mass
and star formation rate surface densities with a slope of $\sim1.1-1.2$, that
is, the so-called KS relation.
