Self-hosting git repos is pretty easy, there are good instructions in the Git book: https://git-scm.com/book/en/v2/Git-on-the-Server-Getting-Git-on-a-Server (excellent book btw, definitely worth a read). There is also a basic web visualiser that comes with Git itself, GitWeb: https://git-scm.com/book/en/v2/Git-on-the-Server-GitWeb
I’ve only ever used it for deployment purposes though (so push / pull etc) hence didn’t bother with a web visualiser. I’m not sure whether it would be suitable for your exact requirements but it would definitely be worth reading through some of those chapters.
I think @NobbZ has worked with a self-hosted set-up with GitLab - hopefully he’ll see this and maybe add some thoughts…