Representation of $x$-non-monotone curves with one intersection each by $x$-monotone curves

Question

Take the $y$-axis and a set of $n$ curves starting from $y$-axis, labelled as $\mathcal{C}:=\{C_1,C_2,...,C_n\}$. These curves fulfill the following conditions:

1. The curves all have a starting point somewhere on the $y$-axis and no two curves share the same starting point.

2. At least one curve is non-$x$-monotone.

3. Any curve intersects with any other curve at most once.

Now, I am thinking of a way to represent the set of curves $\mathcal{C}$ with a set of $x$-monotone curves, say $\mathcal{C}'=\{C_{1}',C_{2}',...,C_{n}'\}$ where the following conditions are fulfilled:

1. All curves in $\mathcal{C}'$ have a starting point on the $y$-axis and no two curves share the same starting point.

2. All curves in $\mathcal{C}'$ are $x$-monotone.

3. Any curve intersects with any other curve at most once.

So I am looking for the following: would there exist a representation of $\mathcal{C}$ with $\mathcal{C}'$ in some way, or perhaps would there exist some example of $\mathcal{C}$ that is not representable by $\mathcal{C}'$? In particular, the order of the intersections of the curves do not matter, just simply the pairs of intersections in $\mathcal{C}$ and $\mathcal{C}'$ must remain the same.

Intuitively, I feel like there should exist some counter example for this, but I am struggling with coming up with such, but it also would seem difficult to prove the given statement.

It seems that maybe the proof of such could be argued with some type of a construction, for instance, labeling the curves from bottom to top as $C_1,....,C_n$ and have $C_1',...,C_n'$ correspond to these curves based on the number of intersection points on each curve. This construction seems unnecessarily complicated and I feel like there could be another way to go about this.

Also something I thought about was obtaining a general construction for this argument through something similar to a wiring diagram (see page 7 here: https://arxiv.org/pdf/1410.2350.pdf). However, I don't think this necessarily works since wiring diagrams are not necessarily $x$-monotone (although some of them can be).

Answer 1

This answer, as pointed out by Jan in the comment, is incorrect, as the definitions slightly differ. I leave it here as it contains useful pointers.

If I understood your definitions correctly, your $\mathcal{C}$ would be called an outerstring graph, and your $\mathcal{C}'$ a double outerstring graph. There are outerstring graphs that are not double outerstring graphs, in fact, even the order of the growth rates of these two families are different. For some literature and a nice summary of results, see the first two pages here: Alexandre Rok and Bartosz Walczak: Outerstring graphs are χ-bounded.