In other words, you want the normalization of $X$ to be smooth, plus a condition on the embedding: the pull back of $\mathcal{O}_X(1)$ to $\tilde{X} $ should be very ample. The latter property depends on the embedding and is not always true, even for curves: see [this post](https://mathoverflow.net/questions/163802/does-normalization-of-projective-varieties-preserve-very-ampleness) and the answers there.