The government need to make more of an effort to fight crimes.
Crime can be countable or uncountable, and as with other nouns that behave like this, the uncountable form has a more general meaning and the countable more specific.
Another way to look at this is to notice that fight and crime (without ‘s’) collocate strongly:
- The government need to make more of an effort to fight crime.
Try googling fight crime and fight crimes. Which is more common? What are the differences in meaning?