Understanding the global carbon cycle requires observation of carbon diffusivity within the convective boundary layer (CBL). In particular, observations of the vertical evolution of CO₂ turbulent flux would enable the evaluation of the spatial representativeness of in situ flux measurements and the quantification of entrainment-flux impacts on such measurements. Coherent lidar is a promising technology for this purpose, as it can perform ground-based range-resolved measurements of CO₂ mixing ratio and vertical wind speed along the height of the CBL. Still, achieving a sufficiently low statistical error in the CO₂ mixing ratio measurement is a key challenge in CO₂ turbulent flux measurements. For this purpose, a 2.05 µm coherent differential absorption lidar (CDIAL) has been developed. This lidar features a hybrid fiber/bulk amplification chain that provides 20 kHz high pulse repetition frequency and 1 mJ pulse energy. These characteristics allow for a random error of about 30 ppm in CO₂ mixing ratio measurements and 0.01 m s^-1 in wind velocity, with a spatiotemporal resolution of 13 s and 115 m, enabling turbulent flux measurements. This paper presents a performance assessment of CO₂ turbulent flux measurements with this CDIAL lidar, under typical meteorological conditions of CO₂ uptake by photosynthesis. A new expression for random error after spatial averaging is introduced, that is verified using Monte Carlo simulations and statistical analysis on experimental measurements. CO₂ fluxes measured by the lidar are discussed with respect to flux measurements made by in situ instruments. The lidar enables flux measurements over the convective boundary layer height (around 1000 m), and achieves a minimum flux detection limit of 11 µmol m^-2 s^-1 at a spatiotemporal resolution of 350 m and 4 hours, making it well suited for future geophysical studies of CO₂ turbulent fluxes.