Electric field effect (EFE) controlled magnetoelectric transport in thin films of undoped and La-doped Sr$_{2}$IrO$_{4}$ (SIO) were investigated under the action of ionic liquid gating. Despite large carrier density modulation, the temperature dependent resistance measurements exhibit insulating behavior in chemically and EFE doped samples with the band filling up to 10%. The ambipolar transport across the Mott gap is demonstrated by EFE tuning of the activation energy. Further, we observe a crossover from a negative magnetoresistance (MR) at high temperatures to positive MR at low temperatures. The crossover temperature was around $sim$80-90 K, irrespective of the filling. This temperature and magnetic field dependent crossover is qualitatively associated with a change in the conduction mechanism from Mott to Coulomb gap mediated variable range hopping (VRH). This explains the origin of robust insulating ground state of SIO in electrical transport studies and highlights the importance of disorder and Coulombic interaction on electrical properties of SIO.
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