The following fact is in the proof of Chevalley's structure theorem on algebraic groups in https://www-fourier.ujf-grenoble.fr/~mbrion/chennai.pdf (Lemma 2.3.5). But I think there is a counterexample. Could you please say is this theorem true or false?:
Let $X$ be a normal variety, $D \subset X$ an irreducible divisor, $Y$ a complete variety, and $f : X \to Y$ a dominant rational map. Then there exist a complete variety $Y′$ and a birational morphism $\phi : Y ′ \to Y$ such that the induced rational map $f′: X \to Y$ restricts to a rational map $f′|D: D \to Y′$ with image $Y′$ or a divisor.
I think this is counterexample:
Let $Y:=\mathbb{A}^2$ and $X$ be a blow up of $Y$ at a point. Then the map $X\to Y\hookrightarrow \mathbb{P}^2$ satisfies the conditions of the theorem but sends the exeptional divisor to a point. So the theorem does not holds for this map.