This is a really awesome project! If you have time/interest, you could try to build a simple http server now, that your browser could communicate with. Then you could try to implement a simple version of TCP/IP, and look into how lower level networking works. Great job!
In my experience, not a good idea to write both the client and the server for a given protocol which use each other. Far too easy to misunderstand the protocol. I remember doing this for a training course I wrote for OLE (later COM) years ago. The client and the server worked perfectly together, just not with correctly implemented OLE clients and servers.