Why not make the touchdown zone of runways wider than the rest?
Because most runways already are as narrow as safely feasible.

Let us look at your question piece by piece:


  Save money by making the rest of the runway narrower


You seem to be assuming we are currently deliberately wasting asphalt on making runways wider than needed, and could build them narrower. As they say on Wikipedia: [citation needed].


  Larger margin of error left and right


The touchdown zone is the whole runway, as long as the aircraft in question can stop afterwards, so if you start down this alley you'll find it cheaper to complete the runway in the same width. 


  Easier to turn around, if necessary


This is already done by including turn pads, on top of the nominal width. For an example, see: OGZ