SQL query to remove certain text from each field in a specific column?

You can use the REPLACE() function in native MySQL to do a simple string replacement.

UPDATE tbl SET file_name = REPLACE(file_name, '.jpg', '');
UPDATE tbl SET file_name = REPLACE(file_name, '.rar', '');

This should work:

UPDATE MyTable
SET file_name = SUBSTRING(file_name,1, CHAR_LENGTH(file_name)-4)