The origin of the hypersurface defined by $$x_0^2 + x_1^4 + x_2^4 + \cdots + x_n^4 = 0$$ is a canonical singularity for $n = 3$, and a terminal singularity for $n \ge 4$ (see e.g. Theorem 2 in this paper). The blowup at the origin is defined by $$x_0^2 + x_1^2(1 + x_2^4 + \cdots x_n^4)$$ in some local chart, which is singular in codimension $1$ and therefore non-normal.

The case $n=3$ is already mentioned in Miles Reid's "Canonical 3-folds".

The origin of the hypersurface defined by $$x_0^2 + x_1^4 + x_2^4 + \cdots + x_n^4 = 0$$ is a canonical singularity for $n = 3$, and a terminal singularity for $n \ge 4$ (see e.g. Theorem 2 in this paper). The blowup at the origin is defined by $$x_0^2 + x_1^2(1 + x_2^4 + \cdots x_n^4) = 0$$ in some local chart, which is singular in codimension $1$ and therefore non-normal.

The case $n=3$ is already mentioned in Miles Reid's "Canonical 3-folds".