In order to get the flag `-depth 4` work, you should use also the flag **`-colors`** with the desired color depth (in your case 16).

The following command should generate an icon with the same size of the original:

    $ convert -flip -colors 16 -depth 4 wikipedia.ico flipped.ico

I have tested it on Windows and the flipped icon has the same size of the original.

See also this [forum](https://www.imagemagick.org/discourse-server/viewtopic.php?t=21456).