Whether this PRG is secure or not
Your intuition that $G'$ is insecure is correct, but your approach is a bit off. In the PRG security game, the adversary is not allowed to influence the choice of seed, which is always chosen uniformly at random.

To attack $G'$, notice that given just $G'(s)$, you can reconstruct $s$ from $\bar{s}$. Therefore a distinguisher, when given a challenge output $w$, can compute $G'(\bar{w}[2n+1] \cdots \bar{w}[3n])$ and check whether this gives $w$.