Let $f,g,h$ be elements in $\mathbb{Z}[x,y]$, each geometrically integral and at least two of them are distinct. Without loss of generality, suppose that $f$ is not proportional to $g$ over $\mathbb{C}$, so that the variety $V_{f,g} = \{f = g = 0\}$ has co-dimension 2 in $\mathbb{A}^2$.
Let $T$ be a large real number, and fix a prime $T/\log T < p < T$. Define the set
$$\displaystyle S_{f,g}(T) = \{\mathbf{x} \in \mathbb{Z}^2 \cap [-T,T]^2 : \mathbf{x} \pmod{\ell} \in V_{f,g}(\mathbb{F}_\ell) \text{ for some } T/\log T < \ell < T \},$$
where $\ell$ runs over primes. One can show that
$$\displaystyle |S_{f,g}(T)| \asymp T.$$
My question is, is it possible for the intersection of $S_{f,g}(T)$ and the set
$$\displaystyle C_h(p; T) = \{\mathbf{x} \in \mathbb{Z}^2 \cap [-T,T]^2 : h(\mathbf{x}) \equiv 0 \pmod{p}\}$$
to be large? Indeed, the question is as in the title, can the "sifted set" $S_{f,g}(T)$ have large intersection with a curve modulo a fixed prime $p$?
