So I'm confused... is Bluetooth turning on when you disable airplane mode or is it just that the icons are visible in the system notification panel of your taskbar and you don't want them there? This is important since they are completely different issues, one being functionality the other being cosmetic.
Did you try turning off Bluetooth prior to enabling airplane mode since airplane mode remembers state of both Bluetooth and wifi? I mentioned this earlier.
If you just don't want to see the icons, you need to customize the notification panel. Right clicking it should show you properties or some similar option to configure that panel. Every desktop environment allows you to edit the panels on the taskbar (goes for windows and Mac as well).