I am trying to find a large subset of piecewise-differentiable plane curves of finite length (subsets of $\mathbb{R}^2$) with the following property:

For any pair $\gamma_1, \gamma_2$ of curves in this class, their images $\Gamma_1, \Gamma_2$ are such that $\Gamma_1\cap \Gamma_2$ has finitely many connected components.

I have attempted to prove that this is the case for this set of curves:

Piecewise-smooth curves (finite length) in which each piecewise component is either a straight line segment or a curve whose derivative is injective,

but I have failed to produce a proof or a counterexample to the claim that this satisfies the desired properties.

Could anybody suggest how to prove it, or why they believe it may be a false claim? If it is false, would some further restriction produce the desired properties?

Obviously, one can restrict to looking at just one piecewise component at a time - and this way it is easy to show that no such curve can intersect a line segment at infinitely many points (using the injectivity of the curve). And of course, any line segment can only intersect another segment at one point or the entire segment. I have failed to produce a proof for when both components are curved. I believe that having infinitely many intersection points should lead to non-injectivity of the curves at some point, but haven't been able to show that.