  We prove that orbit equivalence relations (ERs, for brevity) of generically
turbulent Polish actions are not Borel reducible to ER s of a family which
includes Polish actions of S_\infty, the group of all permutations of N, and is
closed under the Fubini product modulo the ideal Fin of all finite sets, and
some other operations. Our second main result shows that T_2, the equivalence
relation called ``the equality of countable sets of the reals'', is not Borel
reducible to another family of ERs which includes continuous actions of Polish
CLI groups, Borel equivalence relations with G_\delta classes, some ideals, and
is closed under the Fubini product over $Fin.$ Both results and their
corollaries extend some earlier irreducibility theorems by Hjorth and Kechris.
