This is explained in the film's official novelisation. The Joker survived the helicopter crash in exactly the same way that Harley did, by jumping from it before it hit the ground.
Then hell plummeted from the sky in the guise of a crippled, out of
control Chinook helicopter that skidded across rooftops and shredded
its way down Ninth Street, only to careen into the bank’s facade,
destroying glass, stone, and mortar before its sturdy all-steel vault
put an abrupt stop to the aircraft.
The copter was on fire. Nothing inside could have survived.
From across the way, on another rooftop, the Joker watched the flames
rising into the sky. He had jumped just in time. If he had waited even
another ten seconds, he would now be little more than ash.
Of course, this utterly ruins the big reveal at the end of the film, which is presumably why they took this scene out of the shooting script.