As we all know majority of the land in Australia are desert
which are not inhabited by human being, so your theory of Australia
being a big country, is also not accurate. And furthermore, do you
really think that criminal do hide in desert? 
Furthermore, having a big country doesnt mean that will result to
higher crime, infact, if people are scattered across the country,
it will be more difficult to commit crime isn't it?
Contrary to what you said, most crime happens in area with higher
population density, (please do check on statistic report) that is
because crimes are usually related to drugs, gangs, prostitution
racism etc.
As we all know australia are known for their liberal approach
towards drugs and I am surprise that you didnt even mentioned this
as one of the reason for the high crime rate in Australia?
Somehow you just remind edme of the 90.5 radio TV commercial about
the tennis coach speaking to his student's father.