Analytical solutions for the drag force can only be found for creeping flows (Stokes' flow), where the flow inertia can be neglected and the flow is symmetric in time, and simple geometries such as a sphere.
For the limiting case of turbulent flows drag highly depends on the precise flow phenomena such as eddies and separation. In analytic calculations highly turbulent flows are generally approximated as inviscid flows, where the friction characterised by the viscosity is neglected, resulting in no drag whatsoever, something termed D'Alembert's paradox: Drag is the result of pressure differences and friction forces due to viscosity. While inertia definitely dominates far from the boundaries, this simplified approach fails to capture the near-wall viscous effects that result in a drag force.
Furthermore as the drag force strongly depends on unsteady effects, most numerical methods will fail to predict it accurately. The current standard in the industry, RANS turbulence models (such as $k-\epsilon$ or $k-\omega$) apply time-averaging and are prone to horrendous errors in determining integral quantities such as drag and lift. More accurate numerical simulations such as large-eddy (LES) and hybrid RANS-LES models, that apply either of the two models depending on the distance to the nearest wall, are still computationally prohibitive in terms of number of mesh cells and time step size (they require huge computational clusters and still will be under-resolved) and thus experiments are still the cheapest and most accurate method to date for determining lift and drag coefficients for most applications. In cases where manufacturing a working prototype might be expensive and slow, such as in the automotive industry, first a virtual prototype and numerical methods will be used to iteratively improve the concept and then verified with actual measurements. If the precise magnitude of the corresponding quantities is secondary, e.g. when comparing different variants, even using cheap computational methods might give valuable information.