Hatcher has the following exercise in chapter $0$:
Show that the space obtained from $S^2$ by attaching $n$ 2 cells along any collection of $n$ circles in $S^2$ is homotopy equivalent to the wedge sum of $n + 1$ 2-spheres.
So, looking at the case when $n=1$, we can "pinch" the attached disk to a point and we get $S^2 \vee S^2$. Following this pinching idea, the claim seems clear when we are attaching our disks across disjoint circles.
However, when the attaching circles intersect, for example with $n=2$, taking both circles as geodesic circles, applying the idea for the disjoint circles seems to produce the wedge sum of four spheres. Can someone point out what is wrong with my thought process?