Which branch of government makes sure the laws are carried out fairly?

None, as far as I know.

Slightly less snarky (but, perhaps, the above has more truth than one would care to admit), it generally falls to the Executive Branch to carry out the operational functions of the government.

The Executive Branch enforces and carries out the laws passed by the Legislative Branch, so the E.B. would be the one in charge of things that you would say makes sure the nation "runs well" - they do police functions, food inspections, transportation projects, etc.