  Given a graph $F$, we consider the problem of determining the densest
possible pseudorandom graph that contains no copy of $F$. We provide an
embedding procedure that improves a general result of Conlon, Fox, and Zhao
which gives an upper bound on the density. In particular, our result implies
that optimally pseudorandom graphs with density greater than $n^{-1/3}$ must
contain a copy of the Peterson graph, while the previous best result gives the
bound $n^{-1/4}$. Moreover, we conjecture that the exponent $1/3$ in our bound
is tight. We also construct the densest known pseudorandom $K_{2,3}$-free
graphs that are also triangle-free. Finally, we obtain the densest known
construction of clique-free pseudorandom graphs due to Bishnoi, Ihringer and
Pepe in a novel way and give a different proof that they have no large clique.
