We have discovered three certain (SAX J1324.5-6313, 2S 1711-339 and SAX J1828.5-1037) and two likely (SAX J1818.7+1424 and SAX J2224.9+5421) new thermonuclear X-ray burst sources with the BeppoSAX Wide Field Cameras, and observed a second burst ever from a sixth one (2S 0918-549). Four of them (excluding 2S 1711-339 and 2S 0918-549) are newly detected X-ray sources from which we observed single bursts, but no persistent emission. We observe the first 11 bursts ever from 2S 1711-339; persistent flux was detected during the first ten bursts, but not around the last burst. A single burst was recently detected from 2S 0918-549 by Jonker et al. (2001); we observe a second burst showing radius expansion, from which a distance of 4.2 kpc is derived. According to theory, bursts from very low flux levels should last <~ 100 s. Such is indeed the case for the last burst from 2S 1711-339, the single burst from SAX J1828.5-1037 and the two bursts from 2S 0918-549, but not for the bursts from SAX J1324.5-6313, SAX J1818.7+1424 and SAX J2224.9+5421. The bursts from the latter sources all last ~ 20 s. We suggest that SAX J1324.5-6313, SAX J1818.7+1424, SAX J1828.5-1037 and SAX J2224.9+5421 are members of the recently proposed class of bursters with distinctively low persistent flux levels, and show that the galactic distribution of this class is compatible with that of the standard low-mass X-ray binaries.