C750 CREW DEPART TMB TO THE W IN IMC AND CONFLICT WITH A TWIN CESSNA A SHORT TIME LATER WITH AN ACR.

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ACFT HAD JUST DEPARTED TMB ENRTE FOR MSY. OUR CLRNC WAS TO CLB TO 2000 FT MSL ON RWY HDG. AFTER LEVELING OFF AT 2000 FT MSL; CTLR ASSIGNED A NEW HDG OF 290 DEGS. A MIN OR 2 HAD GONE BY WHEN THE CTLR ISSUED US AN IMMEDIATE R TURN TO 020 DEGS. AS WE WERE IN THE TURN; OUR TCASII ALERT SOUNDED AND A 300 SERIES CESSNA COULD BE SEEN APPROX 1200 FT OFF OUR L SIDE EXECUTING A TIGHT R TURN. A FEW MORE MINS LATER; STILL AT 2000 FT MSL AND ON OUR HDG OF 020 DEGS; WE RECEIVED ANOTHER TCASII ALERT FOR TFC LESS THAN 1 MI AT OUR SAME ALT AND 12 O'CLOCK POS. WE INITIATED AN IMMEDIATE CLB JUST AS THE CTLR ISSUED US AN IMMEDIATE CLB TO 6000 FT MSL. WE FEEL THE PROB AROSE DUE TO A HVY TFC AREA; LACK OF CTLR SUPERVISION AND EXCESSIVE WORKLOAD. OUR ACTIONS WERE FOR THE SAFETY OF THE FLT. THE MIAMI CTL FACILITY HAS BEEN NOTIFIED AND HAS EXPLAINED THAT THIS INCIDENT WAS AN OPERATIONAL CTL ERROR. CALLBACK CONVERSATION WITH RPTR REVEALED THE FOLLOWING INFO: RPTR SAID THEY HAD DEPARTED TO W INTO IMC AND CONFLICTED WITH A TWIN CESSNA OBSERVED ON TCASII 1200 FT AWAY INDICATING THE SAME ALT. A SHORT TIME LATER THEY CONFLICTED WITH AN ACR ABOUT AS CLOSE. HE SAID THE CTLR WAS VERY BUSY AND MAY HAVE MISSED ISSUING A CLB CLRNC. HE SAID HIS COMPANY WAS TOLD BY THE MIA CTL FACILITY THAT THE CONFLICT WAS CLASSIFIED AS OPERATIONAL CTL ERROR.