  We consider first-passage percolation on $\mathbb Z^2$ with independent and
identically distributed weights whose common distribution is absolutely
continuous with a finite exponential moment. Under the assumption that the
limit shape has more than 32 extreme points, we prove that geodesics with
nearby starting and ending points have significant overlap, coalescing on all
but small portions near their endpoints. The statement is quantified, with
power-law dependence of the involved quantities on the length of the geodesics.
  The result leads to a quantitative resolution of the
Benjamini--Kalai--Schramm midpoint problem. It is shown that the probability
that the geodesic between two given points passes through a given edge is
smaller than a power of the distance between the points and the edge.
  We further prove that the limit shape assumption is satisfied for a specific
family of distributions.
  Lastly, related to the 1965 Hammersley--Welsh highways and byways problem, we
prove that the expected fraction of the square $\{-n,\dots ,n\}^2$ which is
covered by infinite geodesics starting at the origin is at most an inverse
power of $n$. This result is obtained without explicit limit shape assumptions.
