DEPARTING BWI; A REGIONAL JET FLT CREW CHALLENGE PCT DEP CTLR TO TURN DUE TO WX; TAKING PRECAUTIONARY WX AVOIDANCE ACTION; INADVERTENTLY CLBING ABOVE ASSIGNED ALT.

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TOOK OFF RWY 28 AT BWI. ATC GAVE US A HDG TOWARD SOME WX AND CLRED US TO 9000 FT. WE MADE THE TURN AND TOLD THEM WE COULDN'T STAY ON THE HDG VERY LONG. SHORTLY LATER; I TOLD ATC WE NEEDED TO TURN L TO AVOID THE WX. THEY RESPONDED 'NOT JUST YET.' WE WAITED A LITTLE LONGER AND TOLD ATC WE HAD TO TURN L TO AVOID WX. HE WAS VERY BUSY AND DIDN'T RESPOND. WE GOT TO THE POINT WHERE WE HAD TO TURN OR WE WOULD FLY INTO THE WX; SO WE TURNED L AND ADVISED ATC OF THE TURN. THEY SAID; 'ROGER; LET ME KNOW WHEN YOU CAN TURN R ON COURSE.' WE WERE EXPERIENCING CONTINUOUS MODERATE TURB FROM SHORTLY AFTER TKOF. WHEN WE GOT TO A HDG AWAY FROM THE WX; I NOTICED THAT WE WERE CLBING THROUGH 9400 FT. I ADVISED THE PF (FO) OF HIS ALT AND HE EXPEDITIOUSLY RETURNED TO 9000 FT. ATC THEN GAVE US A CLB CLRNC TO A HIGHER ALT. ATC NEVER DID QUESTION THE ALTDEV. SUPPLEMENTAL INFO FROM ACN 610629: DEPARTED BWI AFTER 2 GND STOPS. WE STARTED TO ENCOUNTER MODERATE TO SEVERE TURB. WE TOLD THEM WE NEEDED TO TURN. THEY STILL SAID WE COULD NOT TURN. WE WERE HEADING RIGHT FOR THE STORM. FINALLY WE AGREED TO TURN TO GET AWAY FROM THE STORM. IN THE MIDST OF FIGHTING WITH ATC AND THE MODERATE TO SEVERE TURB OF THE STORM; I MISSED MY ALT OF 9000 FT BY 500 FT.