How do I create posts that include a picture, but doesn't include that picture in the media section?

So, basically what I want to do is two things;
1) I want to post links where I can choose a picture to represent that article or whatever I am posting e.g. post a tweet that includes a link and then a picture also to represent that link.
2) How do I then avoid the posted picture from showing up and storing in what I refer to as 'the media box' on the twitter page...i.e. the area where you can view someones media they've posted = music, videos, pictures etc.
My main issue is; I want people to only see the music videos I have posted on my twitter and NOT the countless throwaway images I post from day to day.
The only way I can think of thus far is to perhaps store images I want to post in the future on an image hosting site, then post the link via bitly or something i.e. some way of posting images without directly posting on twitter...if you follow.

Any help and advice is greatly appreciated. Thanks.