This work investigates the mechanisms that underlie transitions to turbulence in a three-dimensional domain in which the variation of flow quantities in the out-of-plane direction is much weaker than any in-plane variation. This is achieved using a model for the quasi-two-dimensional magnetohydrodynamic flow in a duct with moving lateral walls and an orthogonal magnetic field, where three dimensionality persists only in regions of asymptotically small thickness. In this environment, conventional subcritical routes to turbulence, which are highly three dimensional (with large variations from nonzero out-of-plane wave numbers), are prohibited. To elucidate the remaining mechanisms involved in quasi-two-dimensional turbulent transitions, the magnetic field strength and degree of antisymmetry in the base flow are varied, the latter via the relative motion of the lateral duct walls. Introduction of any amount of antisymmetry to the base flow drives the critical Reynolds number infinite, as the Tollmien-Schlichting instabilities take on opposite signs of rotation and destructively interfere. However, an increasing magnetic field strength isolates the instabilities, which, without interaction, permits finite critical Reynolds numbers. The transient growth obtained by similar Tollmien-Schlichting wave perturbations only mildly depends on the base flow, with negligible differences in growth rate for friction parameters H≳30. Weakly nonlinear analysis determines the local bifurcation type, which is always subcritical at the critical point, and along the entire neutral curve just before the magnetic field strength becomes too low to maintain finite critical Reynolds numbers. Direct numerical simulations, initiated with random noise, indicate that a subcritical bifurcation is difficult to achieve in practice, with only supercritical behavior observed. For H≤1, supercritical exponential growth leads to saturation but not turbulence. For higher 3≤H≤10, a turbulent transition occurs and is maintained at H=10. For H≥30, the turbulent transition still occurs, but is short lived, as the turbulent state quickly collapses. In addition, for H≥3, an inertial subrange is identified, with the perturbation energy exhibiting a -5/3 power law dependence on wave number.