How to restart TortoiseGit?
Just after posting the question I realized I can try restarting Explorer instead of TG. So, I opened Process Manager and closed the explorer.exe processes running there (the executable of Windows Explorer). Then, I selected File->Run... and typed explorer.exe to re-run the process.
This fixed the problem!
You can try ShellExView to disable and then re-enable TortoiseGit shell extension. You can also Restart Explorer (Ctrl+E) from ShellExView -> Options.