Image of a rational map given by a complete linear system Let $X$ be a smooth projective variety, and $\phi: X\dashrightarrow \mathbb P^n$ a rational map given by some complete linear system $|D|$ where $D$ is a divisor. Is the image variety $\phi(X)\subset \mathbb P^n$ normal?

Not necessarily.
For example, take a general curve $C$ over $\mathbb{C}$ of genus not a triangular number, and $P\in C$ a point.  Let $D=nP$ with $h^0(D)=3$.  Then $\phi\colon C\dashrightarrow\mathbb{P}^2$ extends to $C\to\mathbb{P}^2$ and obviously $\phi(C)$ cannot be smooth (since it would violate the degree-genus formula).
