I'm probably missing the obvious but it`s getting annoying trying to find how to withdraw an injured horse or a horse bought at the sales that already has an entry, only to keep doing the same 10+ minute attempt and failing. Didn`t play SO6 much so it`s been a while since I played.

S06 - click on the horse, then click on it's next start and then you get the option to remove jockey and horse.
I'll double check that to make sure I am right.
Hopefully the same for S07.

There are a couple ways you can do this but the easiest way is by clicking on the horses name so you enter the profile, you'll see by the main name at the top you'll see a little white notepad. Click on this and it'll take you to the advance declarations screen where you can select your race from the drop-down on the right and click 'Remove Horse'

Hope this helps.