I still find it a little dubious.

The thing that makes the site behave slowly is not your computer. The problem is their servers cannot handle the volume of traffic they're receiving, or at least that the volume is creating problems. There's not a lot you can do on your end, though requesting JUST the specific image instead of an entire page might help.

I don't think there's anything about the classes on your end that is going to have an impact. Managing cookies is possible with each type, but it gets harder or easier depending on which class you use. The WebRequest/Response types have the most obvious way to mess with cookies, but I'm pretty sure HttpClient and WebClient have ways too.