Solution 1:

You can make use of wildcards:

DEL "C:\adjust\path\shortcut_name.*"

Bare in mind: this is just a workaround and will delete every file with the same name in the target directory!

Solution 2:

Though it's not shown, even if you have set not to hide extensions in Explorer, all link files have the extension .lnk.

If you create a shortcut to Notepad that appears to be notepad.exe, to delete it from a command prompt, enter:

del notepad.exe.lnk