This may help as a primer:

• https://www.academia.edu/41846183/Handbook_of_Statistical_Distributions_with_Applications
• https://inspirehep.net/files/1ab434101d8a444500856db124098f9c
• https://personal.utdallas.edu/~scniu/OPRE-6301/documents/Important_Probability_Distributions.pdf
• https://en.wikipedia.org/wiki/List_of_probability_distributions

A couple of points - you write: "Even though the sample has been increased further, it is still found that the residuals are not normally distributed". There is no expectation that this will happen - the central limit theorem for instance applies to averages from samples not to the raw data or residuals of a model.

If the residuals are non-normal then the inferences from that model that are based on known distributions that assume normality of residuals will be innacurate. So if you are using t or F distributions for test statistics and CIs then p values and CIs will be wrong. This may not matter much if the residuals are mildly non-normal and the inferences very clear cut, but they should generally be interpreted more cautiously.

Solutions depend on many factors - one is just to be more cautious in interpretation. Another is to look at transformation. Another is to use something like boostrapping to get better inferences. Yet another is to switch to a different model (which weaker assumptions).

Hello Fransiska Soejono. How did you assess normality of the residuals? Thanks for clarifying.

You might read about transformations that normalize v B v

It's no problem for the regression at all. Only your p values might not be valid. They are not really trustworthy anyway.

en.wikipedia.org/wiki/Simple_linear_regression#Normality_assumption

One possibility is the model for the data is incomplete.

If you find evidence of non-normality of residuals, you need to investigate the possible causes and remedies. Common causes include a misspecified model, data containing outliers or influential points, and response or explanatory variables that are skewed or have heavy tails.