Removing outliers can cause your data to become more normal but contrary to what is sometimes perceived, outlier removal is subjective, there is no real objective way of removing outliers.
The problem, as always, is what the heck does one mean by 'outlier' in these
contexts. Seems to be like pornography -- "I know it when I see it."
-- Berton Gunter (quoting Justice Potter Stewart in a discussion about tests
R-help (April 2005)
Always remember that these points remain observations and you should not just throw them out on a whim. Instead you should have good reasons to remove your outliers. There may be many truly valid reasons to remove data-points. These include outliers caused by measurement errors, incorrectly entered data-points or impossible values in real life. If you feel that any outlier are erroneous data points and you can validate this, then you should feel free to remove them.
On the other hand, if you see no reason why your outliers are erroneous measurements then there is no truly objective way to remove them. They are true observations and you may have to consider that the assumptions of your test do not correspond to the reality of your situation. You could always try a non-parametric test (which in general are less sensitive to outliers) or some other analysis that does not require the assumption that your data is normally distributed.