You would probably have to do a piece-wise interpolation. The problem is that your curve goes back on itself rather than progressing monotonically along the x axis. It is in these places that it introduces the spikes.
If you break your curve up into segments each of which is monotonic along x and interpolate each of those and stitch them back together it should work better I think.
How to break up your curve in such a way is something you would need to think about though. It should be easy enough to locate the points at which the x-values change direction I would think, but is beyond the time I have available to suggest in more detail.
