Best statistical test to compare two groups when they have different distributions I am teaching myself to use and apply statistics to a big database.

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*I have 2 groups that I wish to compare, healty controls (HC) and patients (P).


*The sizes are HC= 84, P= 196.


*Each group has many different clinical data collected as continous variables, such as weight, BMI, size of theire frontal lobe, etc.
Given the central limit theorem, I would have thought that I could just assume normality. I corroborated this graphically with histograms, and noticed that in many cases the distribution was not normal. So I performed a shapiro-wilk test, confirming that several variables of both patients and controls had non-normal distribution.
This outcome made me more confused, because now I would like to compare the 2 groups, but they have different distributions.
How should I statistically approch this?
For example, I would like to test if the size of the frontal lobe is statistically different in patients than in controls. BUT the distributions looks like this:

Thanks!

One option would be to use a two-sample Kolmogorov-Smirnov test:
https://en.wikipedia.org/wiki/Kolmogorov%E2%80%93Smirnov_test
This compares the empirical CDFs of the distribution, and computes a test statistic based on the quantile based on the largest discrepancy between the two. The KS test does not require any assumptions about the distributions that the two samples follow. With a large enough sample this test can state whether there is an indication of them following different distributions, though it will not quantity how they are different.
