Magit is not responsible for the version control information that is
being displayed in the mode-line and looks something like Git-master.
The built-in "Version Control" package, also known as "VC", updates
that information, and can be told to do so more often:
(setq auto-revert-check-vc-info t)
But doing so isn’t good for performance. For more (overly optimistic) information see (emacs)VC Mode Line.
If you don’t really care about seeing this information in the mode-line, but just don’t want to see incorrect information, then consider simply not displaying it in the mode-line:
(setq-default mode-line-format
(delete '(vc-mode vc-mode) mode-line-format))