The question is related to question here but with a little of modification which makes the task a little difficult
Let $f:\mathbb{R} \to \mathbb{R}$ and $g: \mathbb{R} \to \mathbb{R}$ be two strictly convex function of classe $C^{1}$ on an open interval $(a,b)$.
Assume $f$ and $g$ are not identical. such that the one-dimensional Hausdorff measure $\mathcal{H}^{1}(\{f(x)=g(x): x\in (a,b)\})=0$
Can we find an upper bound on how many times these two functions can intersect on $(a,b)$?
No, there can be arbitrarily many intersections. In fact there can be infinitely many. Consider the function $h(x)=x^5\sin(\frac{1}{x})$ in $[-1,1]$ (in $0$ we define $h(0)=0$). That function can be checked to be $C^1$ and is $0$ in only countably many points.
Now consider the functions $f(x)=100x^2$ and $g(x)=100x^2+h(x)$ in $(-1,1)$. It is easy to see that their second derivatives are positive in the whole interval, and the two functions are equal in countably many points (so an infinite set but of measure $0$). So that´s a counterexample to the conjecture.