Are you dragging-and-dropping, or doing right-click/Copy, right-click/Paste? Can you open the file on the external drive after you've copied it but before you reboot or remove/replug the drive?
Can you create new files on the external drive (right-click in the Explorer windows, choose New/Text Document). When (if) the New Text Document appears, type something into it using Notepad, save it, re-open it to check it.
Open a command window as admin. Start Menu/Accessories/Command Prompt, right click and choose "Run as Administrator". In the command window type
chkdsk -your external drive letter-: /r /f
chkdsk e: /r /f
It might take some time to complete a thorough disc check. You may need to reboot the PC first.