Context : Near-IR polarimetric images of protoplanetary disks enable us to characterize substructures that might be due to the interaction with ( forming ) planets .
The available census is strongly biased toward massive disks around old stars , however .

Aims : The DARTTS program aims at alleviating this bias by imaging a large number of T Tauri stars with diverse properties .

Methods : DARTTS-S employs VLT/SPHERE to image the polarized scattered light from disks .
In parallel , DARTTS-A provides ALMA images of the same targets for a comparison of different dust components .
In this work , we present new SPHERE images of 21 circumstellar disks , which is the largest sample released to date .
We also recalculated some relevant stellar and disk properties following Gaia DR2 .

Results : The targets of this work are significantly younger than those published thus far with polarimetric near-IR ( NIR ) imaging .
Scattered light is unambiguously resolved in 11 targets , and some polarized unresolved signal is detected in 3 additional sources .
Some disk substructures are detected .
However , the paucity of spirals and shadows from this sample reinforces the trend according to which these NIR features are associated with Herbig stars , either because they are older or more massive .
Furthermore , disk rings that are apparent in ALMA observations of some targets do not appear to have corresponding detections with SPHERE .
Inner cavities larger than \sim 15 au are also absent from our images , even though they are expected from the spectral energy distribution .
On the other hand , 3 objects show extended filaments at larger scale that are indicative of strong interaction with the surrounding medium .
All but one of the undetected disks are best explained by their limited size ( \lesssim 20 au ) , and the high occurrence of stellar companions in these sources suggests an important role in limiting the disk size .
One undetected disk is massive and very large at millimeter wavelengths , implying that it is self-shadowed in the NIR .

Conclusions : This work paves the way toward a more complete and less biased sample of scattered-light observations , which is required to interpret how disk features evolve throughout the disk lifetime .