Electric field effect (EFE) controlled magnetoelectric transport in thin films of undoped and La-doped Sr2IrO4 (SIO) is investigated using ionic liquid gating. 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 channel resistance and chemical doping. We observe a crossover from high temperature negative to low temperature positive magnetoresistance around ∼80-90 K, irrespective of the filling. This temperature and magnetic field dependent crossover is discussed in the light of conduction mechanisms of SIO, especially variable range hopping (VRH), and its relevance to the insulating ground state of SIO.