I was going to recommend imgur
Photobucket works well.
Been using this for 7 years now,not had one problem.
Buy a hosting package, £15 per year from TSOhost, you can put absolutely anything you like on it.
My own one is better than all.Upload, shows you links, done. One button, no ads, fast, etc.
imgur, then use the imgur extension for chrome by metronomik. perfection.
How do you set these up?
I have a hosting package from Vidahost with some websites on it but nothing else, I'm probably very naive about all that I can do with the service.
Is it still as anonymous as, say, an Imgur account?
You don't really need to set anything up, just copy the image files to the servers public html directory and then your link to the images will be domainname/image.jpg
As for anonymous I'm not sure what you mean.