I'm not clear what you mean about turbulence? Quantum gravity involves as-yet unknown physics but turbulence is surely just a case of "the number of calculations we need to simulate it grows quicker than we can reasonably keep up with"? i.e. simple toy simulations work fine but we don't have a planet-sized computer to do something more extensive.
(Just to clarify - this isn't a "classical vs quantum" thing - it's a "we know the equations vs we don't" thing. I'm sure QED simulations are fairly similar within a known domain)
Presumably the GP refers to turbulence being infamously difficult to model analytically. But we have a great model for turbulent flow, the Navier–Stokes differential
equations, it’s just that we can only solve them numerically in all but the simplest of cases. But when we do we get good results, so it’s not like turbulence is some mystical phenomenon beyond the reach of our standard mathematical tools!
The alternative is numerical approximation, as I mentioned. Like always in physics, any analytic (ie. "closed form") solutions would also necessarily have to be special-cases, approximations, and simplifications. The problem with turbulence is that it's so chaotic and complex that it doesn't seem to be reducible to simpler models while still retaining some predictive power. But that's not very surprising; we're talking about the chaotic motion of molecules at the Avogadro scale! After all, we can't even write closed-form solutions (or even approximations) to the motion of merely three bodies under gravity, never mind ten to the power of twenty-three.
But what is fascinating about turbulence, though, is the "edge of chaos" – the boundary conditions at which laminar (non-turbulent) flow suddenly turns turbulent. On one side of the boundary analytic treatment is possible, on the other it is not.
(Just to clarify - this isn't a "classical vs quantum" thing - it's a "we know the equations vs we don't" thing. I'm sure QED simulations are fairly similar within a known domain)