= Solution
The identification with $\mathbf P^1$ means that $C$ is a <smooth plane conic>. An arbitrary singular or nonreduced conic over an <algebraically closed field> cannot be identified with the projective line, so this is a necessary qualification implicit in the requested description.
Choose an equation $\ell=0$ for $L\subset\mathbf P^3$ and a degree-two equation $q=0$ whose restriction to $L$ defines $C$. Its <homogeneous ideal> in $\mathbf P^3$ is $(\ell,q)$. These equations form a <regular sequence>: $\ell$ is a non-zero-divisor in the polynomial ring, and modulo $\ell$ the ring is a polynomial ring in three variables, in which the nonzero quadratic $q$ is again a non-zero-divisor. The generator classes therefore form a basis of the <conormal sheaf>, with their grading shifts:
$$
\mathcal I_C/\mathcal I_C^2\cong\mathcal O_C(-1)\oplus\mathcal O_C(-2).
$$
One can verify the absence of relations from the two-generator <Koszul complex>: a relation between $\ell,q$ is a multiple of $(q,-\ell)$, whose coefficients become zero modulo $\mathcal I_C$. This is the <normal sheaf of a projective complete intersection> calculation. Dualizing the conormal sheaf gives
$$
N_{C/\mathbf P^3}\cong\mathcal O_C(1)\oplus\mathcal O_C(2).
$$
These twists are restrictions from $\mathbf P^3$, not intrinsic degree-one twists on $\mathbf P^1$. A line in the plane cuts the degree-two curve in two points counted with multiplicity, so the <degree of a line bundle> $\mathcal O_C(1)$ is two. The <Picard group of the projective line> then gives
$$
\mathcal O_C(1)\cong\mathcal O_{\mathbf P^1}(2),\qquad\mathcal O_C(2)\cong\mathcal O_{\mathbf P^1}(4).
$$
Equivalently, the degree-two <Veronese embedding> $[s:t]\mapsto[s^2:st:t^2]$ pulls each linear coordinate back to a quadratic. Thus the <normal bundle of a smooth plane conic> is
$$
\boxed{N_{C/\mathbf P^3}\cong\mathcal O_{\mathbf P^1}(4)\oplus\mathcal O_{\mathbf P^1}(2),\qquad\{a,b\}=\{4,2\}.}
$$
The same splitting follows from the normal sequence for the two embeddings:
$$
0\longrightarrow N_{C/L}\longrightarrow N_{C/\mathbf P^3}\longrightarrow N_{L/\mathbf P^3}|_C\longrightarrow0.
$$
Its outer terms are $\mathcal O_{\mathbf P^1}(4)$ and $\mathcal O_{\mathbf P^1}(2)$. The <extension group> class lies in $\operatorname{Ext}^1(\mathcal O(2),\mathcal O(4))=H^1(\mathbf P^1,\mathcal O(2))=0$, by the preceding <Čech cohomology> calculation, so the sequence splits. Both methods distinguish the ambient twist from the intrinsic degree on the curve.
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