Large-area thin films of topological Dirac semimetal Na3Bi are grown on amorphous SiO2:Si substrates to realize a field-effect transistor with the doped Si acting as a back gate. As-grown films show charge carrier mobilities exceeding 7 000cm2/V s and carrier densities below 3×1018cm-3, comparable to the best thin-film Na3Bi. An ambipolar field effect and minimum conductivity are observed, characteristic of Dirac electronic systems. The results are quantitatively understood within a model of disorder-induced charge inhomogeneity in topological Dirac semimetals. The hole mobility is significantly larger than the electron mobility in Na3Bi which we ascribe to the inverted band structure. When present, these holes dominate the transport properties.