We present observations of three new sources in the European Southern Observatory VLT/X-shooter survey dedicated to the detection of the emitting counterparts of damped Ly... (DLA) systems towards bright quasars (QSOs). The aim is to bridge the observational gap between absorption (i.e. DLAs) and emission-selected galaxies at z ~ 2.2-2.5, in order to get a more complete picture of (proto)galaxies around this epoch. The hypothesis is that because DLA galaxies fulfil metallicity-velocity width and luminosity-metallicity relations, high-metallicity DLAs are more likely to be detected in emission. The region around each QSO is covered with slits (1.3 arcsec x 11 arcsec) at three different position angles. In the DLA towards QSO J205922.4-052842 (...), Ly... emission is detected at 3... confidence limit at an impact parameter of <6.3 kpc, and indicates a star formation rate ... for the associated DLA galaxy. We do not detect the associated emission of two other DLAs in the spectra of QSOs J003034.4-512946 (zDLA = 2.452, Zn/H = -1.48 plus or minus 0.34) and J105744.5+062914 (zDLA = 2.499, Zn/H = -0.24 plus or minus 0.11, S/H = -0.15 plus or minus 0.06). We conclude that focusing on metal-rich DLAs is a good way to find counterparts, but the non-detections at high metallicity (e.g. that of the DLA in J105744.5+062914) show that there is not a one-to-one relationship, and cautions us to not naively apply the properties of the DLA counterparts to all metal-rich DLAs. (ProQuest: ... denotes formulae/symbols omitted.)