Do I need to apply for a so-called transit visa to transfer in Singapore and fly all the way to Singapore?

You need a visa! But by the way, do you have an Australian study visa, as long as it is valid? You can enter Singapore with your study visa. See my explanation below.

If you follow the general procedure, you can't take Jetstar back to China, because Jetstar has no flights to the mainland and no partners to fly to the mainland, so when you arrive in Singapore, you must get your luggage through the customs and then take the follow-up flight. (cheap travel to cheap travel, student visa ... this is the first time I have seen you return to China. )

But!

If you have valid visas from the United States, Canada, Britain, France and Australia, you can show your ticket back to Chinese mainland to the Immigration Bureau when you arrive in Singapore (you must go back to the mainland, because I have personally experienced that Singapore is not allowed to enter the country, and the whole journey was signed by Singapore Airlines. I wanted to spend the night in Singapore, but I returned to Hong Kong with a China passport. As a result, I met X's immigration officer, who insisted that I wouldn't go back to Chinese mainland and asked me to show my ticket to the mainland, so that I wouldn't have to stay at the airport for one night.

PS, what do you do? Why go back to China by low-cost airlines? If you are a student, your baggage fee is very expensive.