  In two-dimensional critical loop models, including the $O(n)$ and Potts
models, the spectrum is exactly known, as are a few structure constants or
ratios thereof. Using numerical conformal bootstrap methods, we study $235$ of
the simplest 4-point structure constants. For each structure constant, we find
an analytic expression as a product of two factors: 1) a universal function of
conformal dimensions, built from Barnes' double Gamma function, and 2) a
polynomial function of loop weights, whose degree obeys a simple upper bound.
We conjecture that all structure constants are of this form.
  For a few 4-point functions, we build corresponding observables in a lattice
loop model. From numerical lattice results, we extract amplitude ratios that
depend neither on the lattice size nor on the lattice coupling. These ratios
agree with the corresponding ratios of 4-point structure constants.
