This is what is currently posted to your question:
Let's say I downloaded a bootable ISO of some Linux distribution, because I want to install Linux later. I can burn that ISO onto my USB flash drive. However the device will keep the iso9660 filesystem, so it won't be writable. I can install Linux, but I cannot do anything else with my USB flash drive.
I am not sure exactly was meant by “burn”. This answer defines “burn” to mean copying the ISO file to a flash drive through the use of the dd command. This command is available from Linux, macOS and Windows.
You posted in a comment that the ISO, used in your answer, was for Linux Mint. I downloaded and burned the Cinnamon Edition of Linux Mint 22 LTS to an USB flash drive. The command I used is given below. I entered the command while booted to the current Debian release of peppermintOS.
sudo dd if=~/Downloads/linuxmint-22-cinamon-64bit.iso bs=1M status=progress of=/dev/sdb
The resultant flash drive is both BIOS and UEFI bootable. Regardless of how you boot the flash drive, a partition, with a writable Ext4 volume, is automatically created in the free space at the end of this drive.
Your post currently asks the following question:
So, is it possible convert the ISO into a "normal" ext4 filesystem, and still keep the USB stick bootable?
Converting is not necessary to solve the problem described in your post. You can solve your problem by booting the flash drive to Mint and then shutting down.
For others reading this answer, you may have a Linux ISO file which does not automatically create a new partition, when “burned” to a USB flash drive and then booted. There is also the possibility that a user would want new partition with a format other than ext4. In these cases, I would refer you to the answer posted by Hannu.