  Let $\mu$ be a planar self-similar measure with similarity dimension
exceeding $1$, satisfying a mild separation condition, and such that the fixed
points of the associated similitudes do not share a common line. Then, we prove
that the orthogonal projections $\pi_{e\sharp}(\mu)$ are absolutely continuous
for all $e \in S^{1} \setminus E$, where the exceptional set $E$ has zero
Hausdorff dimension. The result is obtained from a more general framework which
applies to certain parametrised families of self-similar measures on the real
line. Our results extend previous work of Shmerkin and Solomyak from 2016,
where it was assumed that the similitudes associated with $\mu$ have a common
contraction ratio.
