I would like to risk an answer that does not use the language of algebraic geometry. For a pair (complex analytic variety $X$; closed anlalitic subvariety $Y$), $U=X\setminus Y$,
there exists a triangulation such that $Y$ is a subcomplex (see, for example Triangulations of algebraic sets - Hironaka 1974, can be found with google books). In other words $X$ is a simplicial complex, and $Y$ is a subcomplex. Now, if $X$ is normal its sinuglarities are in real codimension at least $4$. I.e. $X$ is a $PL$ manifold in codimension $4$.

In order to show that the fundamental group of $X\setminus Y$ surjects onto the fundamental group of $X$, it is sufficient to show that every loop in $X$ can be homothoped into $X\setminus Y$. Since $Y$ it is contained in the simplicial subcomplex of codimesnion $2$ it is enougth to show that any loop in $X$ can be homothoped so it does not touch any simplex of codim $2$, but this is true for every $PL$ space that is a manifold in codim $2$.

I would like to risk an answer that does not use the language of algebraic geometry. For a pair (complex analytic variety $X$; closed analytic subvariety $Y$), $U=X\setminus Y$,
there exists a triangulation such that $Y$ is a subcomplex (see, for example Triangulations of algebraic sets - Hironaka 1974, can be found with google books). In other words $X$ is a simplicial complex, and $Y$ is a subcomplex. Now, if $X$ is normal its singularities are in real codimension at least $4$. I.e. $X$ is a $PL$ manifold in codimension $4$.

In order to show that the fundamental group of $X\setminus Y$ surjects onto the fundamental group of $X$, it is sufficient to show that every loop in $X$ can be homotoped into $X\setminus Y$. Since $Y$ it is contained in the simplicial subcomplex of codimension $2$ it is enough to show that any loop in $X$ can be homotoped so it does not touch any simplex of codim $2$, but this is true for every $PL$ space that is a manifold in codim $2$.