  The streaming instability (SI) is a leading candidate for planetesimal
formation, which can concentrate solids through two-way aerodynamic
interactions with the gas. The resulting concentrations can become sufficiently
dense to collapse under particle self-gravity, forming planetesimals. Previous
studies have carried out large parameter surveys to establish the critical
particle to gas surface density ratio ($Z$), above which SI-induced
concentration triggers planetesimal formation. The threshold $Z$ depends on the
dimensionless stopping time ($\tau_s$, a proxy for dust size). However, these
studies neglected both particle self-gravity and external turbulence. Here, we
perform 3D stratified shearing box simulations with both particle self-gravity
and turbulent forcing, which we characterize via $\alpha_D$ that measures
turbulent diffusion. We find that forced turbulence, at amplitudes plausibly
present in some protoplanetary disks, can increase the threshold $Z$ by up to
an order of magnitude. For example, for $\tau_s = 0.01$, planetesimal formation
occurs when $Z \gtrsim 0.06$, $\gtrsim 0.1$, and $\gtrsim 0.2$ at $\alpha_D =
10^{-4}$, $10^{-3.5}$, and $10^{-3}$, respectively. We provide a single fit to
the critical $Z$ as a function of $\alpha_D$ and $\tau_s$ required for the SI
to work (though limited to the range $\tau_s = 0.01$--0.1). Our simulations
also show that planetesimal formation requires a mid-plane particle-to-gas
density ratio that exceeds unity, with the critical value being independent of
$\alpha_D$. Finally, we provide the estimation of particle scale height that
accounts for both particle feedback and external turbulence.
