Currently the moon I am seeing over Melboune, Australia is a crescent moon. But it appears to be the bottom of the moon. I only ever remember seeing the left side or right side of the moon as the crescent. So is this an anomaly?
It depends on your latitude. Close the the equator, the more horizontal it looks. Here is a good explanation.
The lit side points toward the sun. If the sun is below the moon, like now, when it’s a very new moon, close to the sun (from our perspective), the lit portion will be pointing down. It’s still the right side, you just have to tilt your head to get the right up-and-down axis.
In the mid-latitudes, the phenomenon you describe is most pronounced near the spring equinox–March for me, September for you. In September the crescent Moon has a much more southerly declination than the Sun, causing it (for you) to be farther from the Sun in altitude but closer in azimuth. For me, in September, the reverse is true; the crescent Moon rides lower along the horizon and the lit side is more rightward.
Cheshire Cat moon.
A waxing moon has the lit side pointing west and a waning moon has the lit side pointing east. Whether it looks like it’s pointing up down or sideways depends on how you’re facing relative to that. That’s pretty much all I’ve retained from astronomy class. 
This page lets you see what the crescent moon looks like in different locations and at different times. Freeze time while the moon is up and you can see how it looks when seen at the same time from different latitudes.
Thank you! That was unexpectedly fascinating.
Crescent moon looks like a toenail.
Actually, if you can mentally draw a line from horn to horn, mentally extend another line perpendicular from the center, and that would tell you where the sun’s position is, assuming it’s below the horizon. But since crescent moons are always closest the sun, that’s pretty typical.
In the same sense, the full moon rises in the east when the sun sets.