Large-area thin films of topological Dirac semimetal Na$_3$Bi are grown on amorphous SiO$_2$:Si substrates to realise a field-effect transistor with the doped Si acting as back gate. As-grown films show charge carrier mobilities exceeding 7,000 cm$^2$/Vs and carrier densities below 3 $times $10$^{18}$ cm$^{-3}$, comparable to the best thin-film Na$_3$Bi. 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. Due to the inverted band structure, the hole mobility is significantly larger than the electron mobility in Na$_3$Bi, and when present, these holes dominate the transport properties.
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