Bremer won three consecutive Danish Formula A karting championships from 1998 to 2000, before racing in British Formula Ford Zetec Championship in 2001. He took one win at Castle Combe and finished second in the prestigious Formula Ford Festival at Brands Hatch driving for Haywood Racing.
For 2002 he entered the British Formula Championship driving for Manor Motorsports. He came second at Knockhill and had a total of 14 top-10 finishes in 26 races, finishing 11th in rookie season.
Sticking with British F3 he finished sixth in the Championship driving for Carlin Motorsports, earning podium results with third place finishes at Snetterton and Rockingham. He ended the season with 10 top-five finishes in 24 starts.
He shifted his career to America in 2004, running in the Toyota Atlantic Series. Racing with Brooks Associates and Polestar Racing, he finished on the podium five times and took victory in Denver after starting fifth. He ended up fifth in the standings and won the BBS of America Rising Star Award.
He was thensigned by the HVM team to drive in Champ Cars in 2005. The rookie was impressive in his first five starts, but he was dropped by HVM for financial reasons, but announced that they had signed Bremer for 2006.
After missing one race, Bremer joined the Dale Coyne Racing team to finish the 2005 season.
Expecting to re-join HVM, now co-owned by Cedric the Entertainer, this proved not to be the case, the team instead signing Dan Clarke and Nelson Philippe.
Bremer returned to Atlantics for the last few races of the 2006 season driving for Polestar. He tested for RuSport in the hope of replacing the injured Cristiano da Matta for the final 2 races of 2006, but the drive went to Ryan Briscoe despite the crash-prone Australian driver being slower. That the first of these 2 races is at Surfer's Paradise in Queensland, and that the race organisers like to have extra Australian drivers for this race, is pure coincidence.