Why don't the Mexicans stay in their own country and fight or protest as they do over here to make their country like the US?

Because many who travel to the US have families back in Mexico. It is much easier an quicker in monetary terms to find a job in the US than to stay and fight the government.

However, many people stay in Mexico and organize countrywide protests against problems such as insecurity or poverty.

Finally, just think about one thing: No matter how many protests were held against George W. Bush, he continued with his own vision of what America should be. If that happens in America, what makes you think such protests would make a difference in a poorer, less democratic country such as Mexico?