In this paper, we study the Hausdorff dimension of self-similar measures and sets on the real line, where the generating iterated function system consists of maps that share a common fixed point. In particular, we show that out of a Hausdorff codimension-one exceptional set of natural parameters, such systems satisfy a weak exponential separation. This significantly strengthens the previous result of Bárány and Szvák [On the dimension of self-similar measures with complicated overlaps. Math. Nachr. 294(4) (2021), 657–671]. As an application, we give the Hausdorff dimension of self-affine measures supported on the generalized 4-corner set.
