Linear Equivalence and Intersection Divisors

Question

Let $C$ be a smooth irreducible projective plane curve and $D$ a divisor on $C$ coming from an intersection: I mean taking for example a Line or a Quadric, intersecting it with $C$ and writing the points of intersection together with the multiplicities according to Bézout's Theorem in the form of a divisor. Let $E$ be a divisor with $E \sim D$ (linear equivalent). Does $E$ also come from an intersection with $C$?