Q: How do I get rid of bought books in iBook?

You can't permanently delete any items from your purchase history, but you should be able to hide them via your computer's iTunes so that they don't show in the Purchased tab for redownloading : http://support.apple.com/kb/HT4919

You can also turn off the showing of books on the bookshelf that are only in the cloud via Settings > iBooks > Show All Purchases 'off'

I've hidden my iBooks which is fine but in Usage it's still showing that iBooks is taking up 1.6GB of space. That is with a completely empty bookshelf. Is that space really gone or is iBooks holding it in reserve in case I redownload everything I've ever acquired? How can I reclaim this space? The iPhone is often telling me I need more Storage.

Thanks.

You no longer have any items (books, PDFs) in the iBooks app ? Only items that are currently downloaded into the app should take up space.

Have you tried closing the iBooks app via the phone's taskbar and doing a soft-reset (i.e. a reboot) of the phone (press and hold both the sleep and home buttons for about 10 to 15 seconds, ignoring the red slider, after which the Apple logo should appear and it should reboot) to see if that frees the space ?

This is not useful because I am still paying for space on my laptop. I want the books to be completely removed.

If you don't want them on your laptop then delete them from it - if the publisher removes them the store then you won't be able to redownload them

I want them permanently deleted, never to be seen from again. Do I have to go to store to get them deleted? Can't I do it online somewhere?

You can't permanently delete items from your purchase history, you can only hide them from the cloud so that they don't show for redownloading : Hide and unhide purchases in iTunes or iBooks on your Mac or PC - Apple Support

As long as they are not downloaded on your laptop then they aren't taking up any space on it, nor are they using any of your iCloud allowance (purchases are separate to your allowance)