Norm.Dist on Test Scores for a Curve

Hello and thank you in advance for your help.

I would like to know the most fair curve possible to give on a test. I want to adjust each individual score in order to make it more "normal" (I don't say normalize because I think that may have a mathematical connotation that doesn't match what I want).

In the attachment, I have on the left some random scores and then the norm.dist function. On the right is the normal distribution. In the middle are two charts: the top is the real scores and the bottom is the normal distribution.

What I want to know is how to adjust each individual score in order to make it more like the normal distribution. I want those at the bottom to get a better curve than those at the very top. Is there some kind of function (or philosophy) I can apply here?

Again, thank you for your help.