Critical behaviour of a system, subjected to strongly anisotropic turbulent mixing, is studied by means of the field theoretic renormalization group. Specifically, relaxational stochastic dynamics of a non-conserved multicomponent order parameter of the Ashkin-Teller-Potts model, coupled to a random velocity field with prescribed statistics, is considered. The velocity is taken Gaussian, white in time, with correlation function of the form $\propto \delta(t-t') /|{\bf k}_{\bot}|^{d-1+\xi}$, where ${\bf k}_{\bot}$ is the component of the wave vector, perpendicular to the distinguished direction ("direction of the flow") --- the $d$-dimensional generalization of the ensemble introduced by Avellaneda and Majda [1990 {\it Commun. Math. Phys.} {\bf 131} 381] within the context of passive scalar advection. This model can describe a rich class of physical situations. It is shown that, depending on the values of parameters that define self-interaction of the order parameter and the relation between the exponent $\xi$