You can use USB-A to USB-C adapters, in either direction, for those things which USB-C have in common: USB data transfers at up to 10 Gbps (USB 3.1 Gen 2 speed).
However, USB-A ports like the ones on your MacBook Air (13-inch, 2017) do not carry video. If you are using an adapter to plug in a standard USB-C monitor or adapter that is looking for a DisplayPort Alt Mode signal, merely "getting the plugs to fit" will not create a video signal out of thin air. That is why your Mac does not even detect your monitor.
A MacBook Air (13-inch, 2017) has an old-style Thunderbolt 2 port, with a Mini DisplayPort connector. This port can act as a Mini DisplayPort, and can support "up to 3840-by-2160 resolution at 60Hz on an external display … at millions of colors."
MacBook Air (13-inch, 2017) - Technical Specifications - Apple Support
If your monitor has a DisplayPort input, or a Mini DisplayPort input, the most straightforward way to connect it to your MacBook Air would be with a Mini DisplayPort to (Mini) DisplayPort cable. If your monitor has a HDMI input, you can get a Mini DisplayPort to HDMI adapter.
If your monitor has a USB-C (DisplayPort) input, I don't believe that you will be able to use it with your MacBook Air. There are plenty of USB-C to Mini DisplayPort adapters, but they are designed to work with the USB-C end plugged into a computer and the Mini DisplayPort end plugged into a monitor – not the reverse. Even if one did work in reverse, it could not carry charging current back to the MacBook Air, or connect monitor USB hub ports. You'll need to wait until you have a newer Mac to make a USB-C monitor connection.