The tight correlation between turbulence and luminosity in giant HII regions (GHRs) is not well understood. While the luminosity is due to the UV radiation from the massive stars in the ionizing clusters, it is not clear what powers the turbulence. Observations of the two prototypical GHRs in the local Universe, 30 Doradus and NGC 604, show that part of the kinetic energy of the nebular gas comes from the combined stellar winds of the most massive stars, the cluster winds, but not all. We present a study of the kinematics of 30 Doradus based on archival VLT FLAMES/GIRAFFE data and new high-resolution observations with HARPS. We find that the nebular structure and kinematics are shaped by a hot cluster wind and not by the stellar winds of individual stars. The cluster wind powers most of the turbulence of the nebular gas, with a small contribution from the combined gravitational potential of stars and gas. We estimate the total mass of 30 Doradus and we argue that the region does not contain significant amounts of neutral (HI) gas, and that the giant molecular cloud 30 Dor-10, which is close to the center of the nebula in projection, is in fact an inflating cloud tens of parsecs away from R136, the core of the ionizing cluster. We rule out a Kolmogorov-like turbulent kinetic energy cascade as the source of supersonic turbulence in GHRs.