My experience of WhatsApp Web has been that the images do not auto-load (Chrome on Windows 10 and Windows 7). A heavily compressed thumbnail is loaded and I have to manually click each image that I want to view.

I haven't analysed any traffic and I'm not a developer but my layman's guess would be that the WhatsApp mobile app has a pre-prepared Zip file (or some other compressed container) with the 20 most recent messages from each chat plus these compressed image thumbnails and the Web Client pulls that data from the phone upon initialising. I doubt it is a big data transfer.