I believe I can make some progress in the real case using a variant of Elekes' approach to the sum-product problem based on the Szemeredi-Trotter theorem for unit paraboloids.
Let $A$ be a finite set of real numbers and $S = \{a^2 : a \in A\}$.
Consider the subset $B \subset \mathbb{R}^2$ given by
$$(a+b, c^2+d^2 : a,b,c,d \in A).$$
Clearly $|B| = |A+A| \times |S+S| $. Now for a given pair $u,v \in \mathbb{R}$ define the paraboloid $p_{(u,v)}$ to be the set of solutions to the equation
$$y= (x-u)^2 + v^2.$$
Now each such paraboloid will contain at least $|A|$ points from $B$, namely the points of the form
$$ (u+a, a^2+v^2)$$
where $a$ varies over $A$ and $u$ and $v$ are fixed (depending on $p_{(u,v)}$). Let $L=\{p_{(u,v)} : u,v \in A \} $ denote the set of parabolas generate as described above, we must have
$$|I(B,L)| \geq |A|^3 $$
On the other hand, by the Szemeredi-Trotter theorem (for "unit" paraboloids), we have
$$|A|^3 \leq |I(B,L)| \lesssim |A+A|^{2/3} |S+S|^{2/3} + |A+A||S+S| + |A|^2$$$$|A|^3 \leq |I(B,L)| \lesssim |A+A|^{2/3} |S+S|^{2/3} |A|^{4/3} + |A+A||S+S| + |A|^2$$
which implies $|A|^{5/2}\lesssim |A+A| \times |S+S|$. This, in turn, implies that
$$|A|^{5/4} \leq \max(|A+A|,|S+S|).$$