How to change the measured curve to be close to the target curve
It is unclear what is your application from your question.

If you aim at "gaussianizing" your data -- for example to make better use of statistical modeling, regression, or machine learning methods which work better under a gaussian assumption -- you can use the Box-Cox transform. It consist in applying the following transformation to your data:

$$y = \frac{x^\lambda - 1}{\lambda}$$

$\lambda$ is chosen as to maximize the log-likelihood of a gaussian model given the transformed data - this is a maximization problem usually solved using Levenberg–Marquardt.

I would also strongly suggest posting this to the stats SE.