This is the main reason why the usual proof of the h-cobordism theorem only works for cobordisms whose boundary has dimension at least 5.
Draw a line on the disc joining two points with the same image.
However this argument seems to be going round in circles: in order to eliminate a double point of the first disc, we need to construct a second embedded disc, whose construction involves exactly the same problem of eliminating double points.
Casson's idea was to iterate this construction an infinite number of times, in the hope that the problems about double points will somehow disappear in the infinite limit.
Informally we can think of this as taking a small neighborhood of the skeleton (thought of as embedded in some 4-manifold).
There are some minor extra subtleties in doing this: we need to keep track of some framings, and intersection points now have an orientation.
Casson handles correspond to rooted trees as above, except that now each vertex has a sign attached to it to indicate the orientation of the double point.
The simplest exotic Casson handle corresponds to the tree which is just a half infinite line of points (with all signs the same).
Freedman's main theorem about Casson handles states that they are all homeomorphic to
as follows from Donaldson's theorem, and there are an uncountable infinite number of different diffeomorphism types of Casson handles.
Freedman's structure theorem can be used to prove the h-cobordism theorem for 5-dimensional topological cobordisms, which in turn implies the 4-dimensional topological Poincaré conjecture.