Command to clear the Git Bash screen

Actually you are looking for a Unix user environment command

clear

or you can use the keyboard shortcut

ctrl+l

http://en.wikipedia.org/wiki/Clear_(Unix)

To clear entire command history in Git Bash.

history -c

try using reset command, it will absolutely clean your screen but you will still have access to previous commands

reset

Neither clear nor history -c does the work actually.
Scroll up, all commands will be visible.

Solution:
If you are in Windows 10, and using mintty 2.7.9 (or above ?) for git bash, use Alt + F8 ... this will work.
Best of luck.
Happy coding.

Reference: here (Perhaps it didn't work for Windows 7)