  We present the $H_{160}$ morphological catalogs for the COSMOS-DASH survey,
the largest area near-IR survey using HST-WFC3 to date. Utilizing the "Drift
And SHift" observing technique for HST-WFC3 imaging, the COSMOS-DASH survey
imaged approximately 0.5 deg$^2$ of the UltraVISTA deep stripes (0.7 deg$^2$
when combined with archival data). Global structural parameters are measured
for 51,586 galaxies within COSMOS-DASH using GALFIT (excluding the CANDELS
area) with detection using a deep multi-band HST image. We recover consistent
results with those from the deeper 3D-HST morphological catalogs, finding that,
in general, sizes and S\'ersic indices of typical galaxies are accurate to
limiting magnitudes of $H_{160}<23$ and $H_{160}<22$ ABmag, respectively. In
size-mass parameter space, galaxies in COSMOS-DASH demonstrate robust
morphological measurements out to $z\sim2$ and down to
$\log(M_\star/M_\odot)\sim9$. With the advantage of the larger area of
COSMOS-DASH, we measure a flattening of the quiescent size-mass relation below
$\log(M_\star/M_\odot)\sim10.5$ that persists out to $z\sim2$. We show that
environment is not the primary driver of this flattening, at least out to
$z=1.2$, whereas internal physical processes may instead govern the structural
evolution.
