Let $C \subset \mathbb{P}^3$ be the twisted cubic, i.e. cut out by the ideal $(wy-x^2,xz-y^2,wz-xy)$. What is a good way to find its conormal sheaf?
It's fairly hard to directly calculate $I/I^2$ as there are so many relations so I doubt that's the right way to proceed...
To compute the conormal sheaf one can use the standard exact seqeunce $$ 0 \to I/I^2 \to \Omega_{\mathbb{P}^3}\vert_C \to \Omega_C \to 0 $$ together with the two Euler sequences $$ 0 \to \Omega_{\mathbb{P}^3} \to \mathcal{O}_{\mathbb{P}^3}(-1)^{\oplus 4} \to \mathcal{O}_{\mathbb{P}^3} \to 0 $$ and (since $C \cong \mathbb{P}^1$) $$ 0 \to \Omega_C \to \mathcal{O}_C(-1)^{\oplus 2} \to \mathcal{O}_C \to 0. $$ Their combination (with an isomorphism $\mathcal{O}_{\mathbb{P}^3}(-1)\vert_C \cong \mathcal{O}_C(-3)$ taken into account) gives an exact sequence $$ 0 \to I/I^2 \to \mathcal{O}_C(-3)^{\oplus 4} \to \mathcal{O}_C(-1)^{\oplus 2} \to 0, $$ where the second arrow is given by the jacobain matrix of the map $$ C \to \mathbb{P}^3,\qquad (u,v) \mapsto (u^3,u^2v,uv^2,v^3). $$ The jacobain matrix is equal to $$ \left(\begin{matrix}3u^2 & 2uv & v^2 & 0 \\ 0 & u^2 & 2uv & 3v^2 \end{matrix}\right) $$ and using it it is easy to check that $$ I/I^2 \cong \mathcal{O}_C(-5)^{\oplus 2}. $$