Turkey hosts almost 3.5M refugees and has to face a humanitarian emergency of unprecedented levels. We use mobile phone data to map the mobility patterns of both Turkish and Syrian refugees, and use these patterns to build data-driven computational models for quantifying the risk of epidemics spreading for measles -- a disease having a satisfactory immunization coverage in Turkey but not in Syria, due to the recent civil war -- while accounting for hypothetical policies to integrate the refugees with the Turkish population. Our results provide quantitative evidence that policies to enhance social integration between refugees and the hosting population would reduce the transmission potential of measles by almost 50%, preventing the onset of widespread large epidemics in the country. Our results suggest that social segregation does not hamper but rather boosts potential outbreaks of measles to a greater extent in Syrian refugees but also in Turkish citizens, although to a lesser extent. This is due to the fact that the high immunization coverage of Turkish citizens can shield Syrian refugees from getting exposed to the infection and this in turn reduces potential sources of infection and spillover of cases among Turkish citizens as well, in a virtuous cycle reminiscent of herd immunity.