Correlation Question

[lisacsa]

I have to do a correlation analysis for a paper and I don't have much experience in practical work.
My data looks like this:

The two variables have a linear (but not monotonic?) correlation but I think it is not normally distributed (last picture) and I wanted to ask if I can still do a Spearman correlation with this distribution?

 

[Etubom]

Conduct data transformation and re-check bow it looks like.