In this work, the interaction of grid-generated turbulence with airfoils of different thicknesses, namely, a National Advisory Committee for Aeronautics (NACA) 0008 and a NACA 0018, is investigated, leading to a deeper understanding of the influence of the airfoil geometry on the near-field flow and on the far-field pressure fluctuations. Experimentally validated lattice-Boltzmann simulations are used to analyze the flow properties in the leading-edge (LE) vicinity. The analysis of the velocity fluctuations near the LE shows that momentum is transferred from the streamwise to the transverse velocity for the NACA 0008 airfoil interacting with a large turbulence length scale. This mechanism changes with the increase in the airfoil thickness because the inflow turbulence length scale becomes comparable to the airfoil thickness in the LE region, resulting in a higher concentration of vortices near the LE oriented in the transverse direction, creating high-velocity fluctuations in the spanwise direction. The near- and far-field pressure fluctuations are analyzed to understand the impact of the inflow turbulence distortion on these parameters and the limitations of analytical methods for real airfoils. Results show that the wall-pressure fluctuations are affected by the turbulence distortion in the LE region. Thick airfoils have noise directivity patterns significantly different compared to the Amiet predictions for higher frequencies, radiating higher noise levels upstream of the LE than the thin airfoil. This is likely associated with a drastic change in the pressure fluctuation distribution near the airfoil LE region, attributed to the change in the distortion of the vortical structures in the LE area.