I\'m writing an app that supposed to copy a bunch of files from one place to another. When I\'m using TFileStream for the copy it is 3-4 times slower than copying the files
Perhaps you can study Cobian Backup 8 (codenamed Black Moon) source code. It is opensource, written in Delphi.
http://www.educ.umu.se/~cobian/cobianbackup.htm