The expansion index and relative contact discontinuity positions of young type Ia supernova remnants (SNRs) evolving in a turbulent medium are investigated via implementing two-dimensional cylindrical magnetohydrodynamic simulations. In our simulations, two kinds of initial ejecta density profiles are considered: a power-law (PL) profile and an exponential (Exp) profile; large-scale density and magnetic field fluctuations are calculated and mapped into the computational domain before the simulations; the back reaction of the accelerated particles is mimicked through a time-dependent effective adiabatic index gamma(eff)(n, t). Our simulation results reveal that the density distributions and magnetic field structures have similar results for both initial profiles. Concerning the expansion index, both the PL and Exp profiles exhibit significant azimuthal variations along the remnant periphery, while the Exp profile manifests itself in a quick path down to the Sedov value as time proceeds. The derived expansion index distributions with both initial profiles are compatible with observations of two typical young type Ia SNRs: SN 1006 and Tycho. As for the relative contact discontinuity positions, when gamma(eff)(n, t) works, the derived profiles of both scenarios are globally compatible with the observations despite some deviations. Besides, there are no obvious ejecta protrusions beyond the forward shock, with a lowest derived ratio reaching similar to 1.01-1.02 for both initial profiles. Moreover, for purposes of comparison, we have also considered two supplementary scenarios: (1) the aforementioned two density profiles expanding in a homogeneous medium, and (2) a uniform ejecta profile evolving in a turbulent medium.