In Ross Barrett’s 1992 novel Suns of Home (published in South Australia) the word broggie appears: ‘Harry races her off in a series of broggies with mum squealing.’ A broggie can be performed with a bike, with a motorcycle, or with a car:

Pine Rd, Woodcroft, will remain a dirt road despite more than 350 people signing a petition calling for it to be bituminised for dust and safety reasons. ... Vines Golf Club of Reynella general manager Paul Sutcliffe said the road was a magnet for hoon drivers. ‘Because it’s not bituminised we have drivers who are spinning wheels and doing broggies, often at night’, he said. (Southern Times Messenger, 9 September 2009)

Parent John Klomp said that some non-skaters were causing problems by driving on the dirt around the park. ‘You get that at any footy oval (next door). I used to happen before the skate park was here they used to do broggies in the dirt.’ (News Review Messenger, Adelaide, 8 August 2001)

I am sure I speak for many of these people when I say that doing burnouts, broggies, donuts, drags and blockies around this suburb at 4am is unacceptable. (Advertiser, Adelaide, 9 January 2006)

The evidence here is from South Australia, but the term is also used in Western Australia and in some other parts of Australia.

The broggie is a sharp turn or slide of a vehicle, with the aim being to throw up a lot of dust. The word is an abbreviation of broadside, with the addition of –i. There is some evidence of a form broadie, so this is probably the transitional form on the way to becoming a broggie. The original broadside in this sense appeared in the 1930s in the form of a verb that meant: ‘Of a motor cycle in dirt-track racing: to skid and slide in the cinders as part of a deliberate manoeuvre when the rider is "hugging" a corner. Also in extended use of a car, pedal cycle, etc., esp. in a controlled sideways skid’ (OED)