There is a way to get over the problem. Create a shortcut to your Picture folder on the Desktop and drag it onto the Start taskbar. url
Find an image on a webpage. Cursor over the pic and drag it onto the shortcut.
Now upload the picture as you normally would by clicking the add icon at the bottom of your new tweet. Choose the image which will be on your computer from the Picture folder and click "Open".
This is far quicker that saving it to the Picture folder then uploading.