We present results from an ongoing X-ray survey of Wolf-Rayet (WR) galaxies, a class of objects believed to be very young starbursts. This paper extends the first X-ray survey of WR galaxies by Stevens & Strickland by studying WR galaxies identified subsequent to the original WR galaxy catalogue of Conti. Out of a sample of 40 new WR galaxies a total of 10 have been observed with the ROSAT PSPC, and of these seven have been detected (NGC 1365, NGC 1569, I Zw 18, NGC 3353, NGC 4449, NGC 5408 and a marginal detection of NGC 2366). Of these, all are dwarf starbursts except for NGC 1365, which is a barred spiral galaxy possibly with an active nucleus. We also report on observations of the related emission-line galaxy IRAS 0833+6517. The X-ray properties of these galaxies are broadly in line with those found for the original sample; they are X-ray overluminous compared with their blue luminosity and have thermal spectra with typically kT~0.4-1.0keV. There are some oddities: NGC 5408 is very overluminous in X-rays, even compared with other WR galaxies; I Zw 18 has a harder X-ray spectrum; NGC 1365, although thought to contain an active nucleus, has X-ray properties that are broadly similar to other WR galaxies, and we suggest that the X-ray emission from NGC 1365 is due to starburst activity. A good correlation between X-ray and blue luminosity is found for the WR galaxy sample as a whole. However, when just dwarf galaxies are considered there is little evidence of correlation. We discuss the implications of these results on our understanding of the X-ray emission from WR galaxies and suggest that the best explanation for the X-ray activity is starburst activity from a young starburst region.