As far as I know (I have pretty good SA) no US airline will sponsor an immigrant. They have enough applicants as it is. That might change if the theoretical pilot shortage materializes here over the next few years...I personally don't believe in pilot shortages any more than the Easter Bunny, but if there was ever a time when the cold, hard numbers support the possibility this is it. Maybe regionals will sponsor visas if there's a pilot shortage.

The good news is that airlines have NO problem hiring anyone holding a green card, in fact there are surprising number of European and Commonwealth pilots at US airlines. So if you can get one of those...

Corporate aviation here is all about who you know. If it's a big enough operation and they like there's nothing to stop them from sponsoring you. But you probably don't know anyone here.

Flight schools will sponsor visas, but their pay and benefits are very poor.

In fact I imagine that you would be shocked at the pay/benefit packages at many US airlines...I would think real hard before leaving Qantas.

Maybe look at Cathay...you might eventually be able to get based in the US on the cargo side.