This simply can not be true.
If the match is on March 15; 4 months would be July 15, allowing not even the most pristine "AMG, with no red flags" to begin.
Needless to say I know some AMG's do not receive their diploma until June regardless.
Yes the process is more annoying for IMG's (waiting for ECFMG certificate (about 2-3 weeks in not expediated), then sending it to the medical state board (electronically), annoying i agree.
But 3-4 months simply not true.
My advice ask your PC/person in charge of credentialing they are usually helpful. The program wants the same thing as you, to start on July 1; they need you to start then just like you need them, case closed.