I will be travelling soon to the US from Jakarta. I have both US and Indonesian passports. Indonesia does not recognise dual citizenship, yet the US requires me to enter and exit the US with my US passport. I've read that I can go to Singapore first and then fly out to the US. But which passports to use and how?
Book two tickets: CGK<>SIN and SIN<>Wherever you're going in the US. Use the Indonesian one for Jakarta<->Singapore. Then use your US passport while entering and leaving the US.
You might want to stop over in Singapore on the way out, and pass through immigration, so that you get a Singapore entry stamp dated the day you left Jakarta. In Singapore, you can leave through the automated immigration channel, even as a tourist, so you don't get a stamp anyway. Indonesian Immigration will see your entry stamp in Singapore, and if they poke around, your return ticket CGK<>SIN. That's it. Keep your US passport well hidden.