Bicuspid aortic valve (BAV), i.e. the fusion of two aortic valve cusps, is the most frequent congenital cardiac malformation. Its progression is often characterized by accelerated leaflet calcification and aortic wall dilation. These processes are likely enhanced by altered biomechanical stimuli, including fluid-dynamic wall shear stresses (WSS) acting on both the aortic wall and the aortic valve. Several studies have proposed the exploitation of 4D-flow magnetic resonance imaging sequences to characterize abnormal in vivo WSS in BAV-affected patients, to support prognosis and timing of intervention. However, current methods fail to quantify WSS peak values. On this basis, we developed two new methods for the improved quantification of in vivo WSS acting on the aortic wall based on 4D-flow data. We tested both methods separately and in combination on synthetic datasets obtained by two computational fluid-dynamics (CFD) models of the aorta with healthy and bicuspid aortic valve. Tests highlighted the need for data spatial resolution at least comparable to current clinical guidelines, the low sensitivity of the methods to data noise, and their capability, when used jointly, to compute more realistic peak WSS values as compared to state-of-the-art methods. The integrated application of the two methods on the real 4D-flow data from a preliminary cohort of three healthy volunteers and three BAV-affected patients confirmed these indications. In particular, quantified WSS peak values were one order of magnitude higher than those reported in previous 4D-flow studies, and much closer to those computed by highly time- and space-resolved CFD simulations.