  We define and study new classes of quasi-hereditary and cellular algebras
which generalize Turner's double algebras. Turner's algebras provide a local
description of blocks of symmetric groups up to derived equivalence. Our
general construction allows one to `schurify' any quasi-hereditary algebra $A$
to obtain a generalized Schur algebra $S^A(n,d)$ which we prove is again
quasi-hereditary if $d\leq n$. We describe decomposition numbers of $S^A(n,d)$
in terms of those of $A$ and the classical Schur algebra $S(n,d)$. In fact, it
is essential to work with quasi-hereditary superalgebras $A$, in which case the
construction of the schurification involves a non-trivial full rank sub-lattice
$T^A_\mathfrak{a}(n,d)\subseteq S^A(n,d)$.
