  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.
