Fundamental understanding of the charge transport physics of hybrid lead halide perovskite semiconductors is important for advancing their use in high-performance optoelectronics. We use field-effect transistors (FETs) to probe the charge transport mechanism in thin films of methylammonium lead iodide (MAPbI$_{3}$). We show that through optimization of thin-film microstructure and source-drain contact modifications, it is possible to significantly minimize instability and hysteresis in FET characteristics and demonstrate an electron field-effect mobility (μ$_{FET}$) of 0.5 cm$^{2}$/Vs at room temperature. Temperature-dependent transport studies revealed a negative coefficient of mobility with three different temperature regimes. On the basis of electrical and spectroscopic studies, we attribute the three different regimes to transport limited by ion migration due to point defects associated with grain boundaries, polarization disorder of the MA$^{+}$ cations, and thermal vibrations of the lead halide inorganic cages. ; S.P.S. acknowledges funding from the Royal Society London for a Newton Fellowship. B.Y. acknowledges support from China Council Scholarship and Cambridge Overseas Trust. A.S. and R.H.F. acknowledge funding and support from the Engineering and Physical Sciences Research Council (EPSRC) through the India-U.K. APEX project. P.D. acknowledges support from the European Union through the award of a Marie Curie Intra-European Fellowship. X.M. is grateful for the support from the Royal Society. B.N. is grateful for the support from Gates Cambridge and the Winton Program for the Physics of Sustainability. We acknowledge funding from the EPSRC through a program grant (EP/M005143/1). We acknowledge funding from the German Federal Ministry of Education and Research under agreement number 01162525/1. This work was performed in part on the SAXS/WAXS beamline of the Australian Synchrotron, Victoria, Australia (55, 56). C.R.M. acknowledges support from the Australian Research Council (DP13012616).