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][1]). 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".


  [1]: https://www.ams.org/journals/tran/2002-354-05/S0002-9947-02-02879-9/S0002-9947-02-02879-9.pdf