Let $X$ be a smooth, complete algebraic variety and suppose I have two projective, birational morphisms

$$f:X \to \mathbb{P}^n$$

and

$$g:X \to \mathbb{P}^m,$$

such that the image of $f$ is the normalization of the image of $g$. Let $L_f$ and $L_g$ the two line bundles on $X$ whose global sections induce respectively $f$ and $g$, in the standard way. Does the condition I require on the normalization of the image imply that $L_f=L_g$?

PS I guess that actually the birationality of the two morphisms is not needed.