I am working with Elementary Juno (4.15.0-43-generic on x86_64). Epiphany does not start anymore in the desktop and I get a segmentation fault while starting in the Terminal.
There's no way to debug a crash without a stack trace, so you should first get a stack trace for the crash, with debug symbols for every frame in the trace. Once you have a stack trace, you can report it to the Epiphany developers. See these instructions for how to acquire a quality stacktrace. Unfortunately, this is a bit harder to do in elementary than in other distros, because you must manually identify and install the required debuginfo packages. You know you've succeeded when there are no more ??? frames in the trace. Please take the backtrace with
bt full rather than normal
Note that this answer applies to any crashing program, not just Epiphany.