There are $3$-folds contractions $f : X \longrightarrow Y$ which contract only a $\mathbb{P}^1$ (set-theoretically) and the conormal bundle of this $\mathbb{P}^1$ is $\mathcal{O}(-1) \oplus \mathcal{O}(3)$.

There are $3$-folds contractions $f : X \longrightarrow Y$ which contract only a $\mathbb{P}^1$ (set-theoretically) and the conormal bundle of this $\mathbb{P}^1$ is $\mathcal{O}(-1) \oplus \mathcal{O}(3)$.

On the other ahnd, this is precisely what locally $\mathbb{Q}$-factorial singularities are done for. To avoid this kind of weird phenomena.