How to run a bash script from a Makefile?

Just like calling any other command from a makefile:

target: prerequisites
    shell_script arg1 arg2 arg3

Regarding your further explanation:

.PHONY: do_script

do_script: 
    shell_script arg1 arg2 arg3

prerequisites: do_script

target: prerequisites 

Each of the actions in the makefile rule is a command that will be executed in a subshell. You need to ensure that each command is independent, since each one will be run inside a separate subshell.

For this reason, you will often see line breaks escaped when the author wants several commands to run in the same subshell:

targetfoo:
        command_the_first foo bar baz
        command_the_second wibble wobble warble
        command_the_third which is rather too long \
            to fit on a single line so \
            intervening line breaks are escaped
        command_the_fourth spam eggs beans