The basic issue is that Australia naive measurements and numbers carry a three tier definition of homelessness: at the lowest level you have what we call "sleeping rough" which is probably the concept closest to what most people and Americans think of as "homeless". But the Australian definition also includes the likes of insecure accommodation and inappropriate accommodation: couch surfers, people living in accommodation with inadequate living conditions, people sleeping in their cars, in socially provisioned homeless accommodation and emergency/crisis accommodation (domestic violence, youth, men and aged issues were the traditional breakdown of most services in my day).
To compare between the two nations from official sources you have to bring them back onto a somewhat comparable basis.
Another source include homeless service provisions, but last time I looked at those they didn't always tend to be on a individual natural person basis.
lastly, homeless service provision, with a few caveats, is one of the only things that might locally and empirically show behaviours of what economists call a "giffen good". And, somewhat paradoxically, provision of homeless services can, to a point, increase the percieved systemic demand for homeless services. increased supply can also create more (real and percieved) demand. the relationship these complexities have with trying to measure homelessness is tricky to say the least...
The obvious alternative explanation is that Americans might simply be less tolerant of measures to decrease the visibility of homeless.