Surjection from a connected separated scheme

Question

Let $S$ be a scheme. Let $X$ be a connected scheme separated over $S$. Let $X\rightarrow Y$ be a surjective $S$-morphism. Is $Y$ separated over $S$? Without connectedness there are counterexamples like the projection from two copies of affine line to the affine line with double origin. What if $X$ is moreover irreducible?

Answer 1

Let $S=\operatorname{Spec}\mathbb{C}$, $X=\mathbb{A}^1$, and $Y$ be $\mathbb{A}^1$ with doubled origin. Then there is a surjective morphism $X\to Y$ given by $x\mapsto (x-1)^2$ which maps $1$ to one of the origins and $-1$ to the other.

Answer 2

In the positive direction, if the surjection is universally closed, then $Y$ is separated over $S$ (https://stacks.math.columbia.edu/tag/09MQ).