Another approach - if the right half is more than half the points - is a robust linear fit, one that can throw out half the data. Indeed if you can bound it smaller than a half you can have more choice about how you do it.
That will literally throw out half the data, but you don't control which half, so if the right side has more curve than the middle it would throw out the right side.
The robust approach is very easy (and it cares less about issues like heteroskedasticity).
If we don't have good ideas about error distributions we can consider something like bootstrapping, if there's enough data.
The more notion you have about the left half, the more likely it is that some additional approaches may be possible.
There's also smoothing, of various kinds. Perhaps a variable-span kernel smoother, where the right side is forced to have a really big span, and the left has a narrower one
Some of these approaches may be adapted to nonlinear functions on the right, some not so much.
I should go, but I can probably talk some more another time. You can always ping me from here and if I am around I'll pop in.