A few years ago this was asked a lot, so added the answer to the FAQ. Copying it here:
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
Version Control and the Mode Line.
If you don’t really care about seeing that information in the mode-line, but just don’t want to see incorrect information, then consider disabling VC when using Git:
(setq vc-handled-backends (delq 'Git vc-handled-backends))
Or to disable it completely:
(setq vc-handled-backends nil)