Fix $m$ points $p_1,\dots,p_m$ in the complex projective space $\mathbb{P}^2$, and assume that they are in general position, i.e. they are all distinct and no three of them lie on the same line, no six of them lie on the same conic, no ten lie on the same cubic, and so on. I am interested in the following problem: for an irreducible curve $C\subset\mathbb{P}^2$, estimate the number $$\sum_{i=1}^m\mathrm{mult}(C,p_i)$$ in terms of the degree $d$ of $C$ and the number $m$ of points. In fact I would be satisfied with just an "asymptotic" estimate for this number when $m$ is very large.
There is an estimate, at least in the case when $2m=s(s+3)$ for some number $s$, so that there is a degree $s$ curve $S$ passing exactly once through all the points. Please tell me if the following is correct: we can apply Bezout to find $$\sum_{i=1}^m\mathrm{mult}(C,p_i)\leq S.C\leq s\,d$$ if $C$ is distinct from $S$ and $$\sum_{i=1}^m\mathrm{mult}(C,p_i)=m$$ if $C$ and $S$ coincide. In both cases, if $s\geq 3$ (so that $m\geq 9$), for any irreducible curve $C$ we find the estimate $$\sum_{i=1}^m\mathrm{mult}(C,p_i)\leq s\cdot\mathrm{deg}(C).$$
As $s=\sqrt{2m+9/4}-3/2$, this estimates the "total multiplicity" of $C$ on $\{p_1,\dots,p_m\}$ to grow as $\sqrt{m}$ for large $m$. Is it possible to do better? Are there well-known estimates for this number?