A simple method, the multiscale minimal Lagrangian map (MMLM) approach, to generate synthetic turbulent vector fields was previously introduced [C. Rosales and C. Meneveau, Phys. Fluids 18, 075104 (2006)]. It was shown that the synthesized fields reproduce many statistical and geometric properties observed in real, isotropic, turbulence. In this paper we investigate if this procedure, which applies a minimal Lagrangian map to deform an initial Gaussian field, can produce also anomalous scaling in the inertial range. It is found that the advection Lagrangian map time scale is crucial in determining anomalous scaling properties. With the sweeping time scale used in the MMLM approach, non-Gaussian statistics and realistic geometric features are reproduced at each scale, but anomalous exponents are not observed; i.e., we observe nearly 1941 Kolmogorov scaling. Conversely, if the appropriate Kolmogorov inertial-range turnover time scale is used in a modified approach [the multiscale turnover Lagrangian map (MTLM) method], fields with realistic anomalous scaling exponents are reproduced. Remarkably, the intermittency and multifractal nature of the energy dissipation is also found to be quite realistic. Finally, the properties of the pressure field derived from the MTLM velocity field are studied and found to be quite realistic also. The results shed new light on what are minimal dynamical requirements for the generation of anomalous scaling and intermittency in turbulent flow: at least one turnover time for small eddies to be sufficiently deformed, as well as the accumulation of spatially correlated deformations across scales.