  We study how the interaction between the streaming instability and intrinsic
gas-phase turbulence affects planetesimal formation via gravitational collapse
in protoplanetary disks. Turbulence impedes the formation of particle clumps by
acting as an effective turbulent diffusivity, but it can also promote
planetesimal formation by concentrating solids, for example in zonal flows. We
quantify the effect of turbulent diffusivity using numerical simulations of the
streaming instability in small local domains, forced with velocity
perturbations that establish approximately Kolmogorov-like turbulence. We find
that planetesimal formation is suppressed by turbulence once velocity
fluctuations exceed $\langle \delta v^2 \rangle \simeq 10^{-3.5} - 10^{-3}
c_s^2$. Turbulence whose strength is just below the threshold reduces the rate
at which solids are bound into clumps. Our results suggest that the
well-established turbulent thickening of the mid-plane solid layer is the
primary mechanism by which turbulence influences planetesimal formation and
that planetesimal formation requires a mid-plane solid-to-gas ratio $\epsilon
\gtrsim 0.5$. We also quantify the initial planetesimal mass function using a
new clump-tracking method to determine each planetesimal mass shortly after
collapse. For models in which planetesimals form, we show that the mass
function is well-described by a broken power law, whose parameters are robust
to the inclusion and strength of imposed turbulence. Turbulence in
protoplanetary disks is likely to substantially exceed the threshold for
planetesimal formation at radii where temperatures $T \gtrsim 10^3 \ {\rm K}$
lead to thermal ionization. Planetesimal formation may therefore be unviable in
the inner disk out to 2-3 times the dust sublimation radius.
