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Self-organized criticality in the Hwa-Kardar model of "running sandpile" [Phys. Rev. A 45, 7002 (1992)] with a turbulent motion of the environment taken into account is studied with the field theoretic renormalization group (RG). The turbulent flow is modelled by the synthetic $d$-dimensional anisotropic velocity ensemble introduced by Avellaneda and Majda [Commun. Math. Phys.131: 381 (1990)] with finite correlation time. The Hwa-Kardar model with time-independent (spatially quenched) random noise is considered alongside the original model with the white noise. The aim of the present paper is to explore fixed points of RG equations which determine the possible types of universality classes (regimes of critical behavior of the system) and critical dimensions of the measurable quantities. Our calculations demonstrate that influence of the type of the random noise is extremely large: in contrast to the case of the white noise where the system possess three fixed points, the case of the spatially quenched noise involves four fixed points with overlapping stability regions. This means that in the latter case the critical behavior of the system depends not only on the global parameters of the system which is the usual case, but also on the initial values of the charges (coupling constants) of the system. These initial conditions determine the specific fixed point which will be reached by RG flow. Since now the critical properties of the system are not defined strictly by its parameters the situation may be interpreted as universality violation. Such systems are not forbidden but they are very rare. It is especially interesting that the same model without turbulent motion of the environment does not predict this nonuniversal behavior and demonstrates the "usual" one with prescribed universality classes instead [J.~Stat. Phys. 178, 392 (2020)].