Let A be an abelian variety over a number field K. The number of torsion points defined over a finite extension L is bounded polynomially in terms of the degree [L:K]. We compute the optimal exponent for this bound, in terms of the dimension of abelian subvarieties and their endomorphism rings for a large class of $A$. More precisely,  we assume that: (1) A is geometrically isogenous to a product of simple abelian varieties of type I, II or III, in the Albert classification; (2) A is of Lefschetz type; (3) A satisfies the Mumford--Tate conjecture. Our result is unconditional for a product of simple abelian varieties of type I, II or III with specific relative dimensions. Further, building on work of Serre, Pink, Banaszak, Gajda and Krason, we also prove the Mumford--Tate conjecture for a few new cases of abelian varieties of type III.