The web server I've written is now serving several websites I'm hosting, some of which have thousands of images, so there's plenty of scope for connection re-use.

Each connection ID is different within dev tools in Chrome - although errors have 0 - (often sequential, sometimes there are huge gaps). And this is the same with HTTP websites on the net (just tried www.briscoes.co.nz and am still seeing the same thing in Chrome 72).

Using Wireshark I can see that Firefox is doing the correct thing, so I'm not sure why you think what I'm looking at is too low-level: I think it's pretty clearly something the client's doing at the TCP level to close the connection its end after its received the content length.

It may well be that I'm not sending a header Chrome's looking for, but I'm not sure what it is, as in its request it's sending "Connection: keep-alive", and the response has the same, and Firefox works fine.

It's almost like it's a client configuration or something, but I don't know what that would be, as I've tried on several machines.