There is a company I'm working with that says we are slowing down their web hosting software by hosting images on a separate domain.

I've told them what we are doing should only speed them up because there will be less file requests to their server.

They replied by saying that because they use HTML 4.0, their server is having to make image requests on the server side before they send content to the user.

This makes no sense to me and am trying to disprove this claim. Am I wrong and just crazy? I've been looking for articles on this for hours and have had no luck.

Proof that their statement is false would be greatly appreciated, and an article on this topic would be even more helpful.

Your mindset is correct. There is nothing about HTML4 that validates their claim in the context you provided us.

When you make a GET request to the server, you pull an HTML page. The browser then parses the document and makes additional requests, as declared in the document. Images are no exception. When it reaches an image, it will make a GET request to retrieve it to the specified URI. If that URI is not on the same domain, it is not going to make a request to the same domain. The server does not make the GET request for you.

Now, they could be doing something special that would cause it to operate more slowly, but nothing about the HTML4 spec would lead to it.