TL;DR: No they shouldn't. You have to push first.
Can you clarify what you mean by your git repo?
Is it a folder on your pc where you have your code and a ".git" folder?
Is it a space on a page like github or git lab?
Also, what does "connected" mean ?
Explanation (based on some guesses on the points above):
If you have just a local folder, then be aware that git is not a "synchronization" tool like dropbox or onedrive, even though visual studio makes it look like so (That's bad design!). There are normally some manual steps to do before you will see your code in azure. And some of those need to be re-done every time you change something (at least stage, commit, push). The references below may help you. Note that your local .git folder is often called the "local repository" and your code your "working copy", while your Azure space is the "remote".
https://learn.microsoft.com/en-us/azure/devops/repos/git/pushing?view=azure-devops&tabs=visual-studio
https://www.tutorialspoint.com/git/git_basic_concepts.htm
https://www.freecodecamp.org/news/learn-the-basics-of-git-in-under-10-minutes-da548267cc91/
Git is difficult in the beginning, but don't let yourself be discouraged, it will get easier!