The link below leads to a clip of passing the first flamethrower in Flying Battery Act 1 with sonic. Its not easy at all and requires very precise timing/inputs. However you can download the video and play it frame by frame so you can see the inputs used in each frame.

https://drive.google.com/file/d/1AzAejxCVmEhNSzT7HY2T3OCMVYFWbrFy/view?usp=sharing

Edit: If you loop jump at the beginning, you will get the best of the flamethrower pass. If you don't loop jump and just run through normally, it is still possible to bypass the flamethrower. However you are better off hitting the wall at the left instead of trying to passing under the wall. Youre gonna get hit doing by the flame if you try to just go under the wall. This is due to the cycle of the flamethrower. Alternatively, you could just get hit by the flamethower. But if you are trying to catch certain cycles for the rest of the stage, you need to be knockbacked at the left angle. If you get knockbacked right, then you are not going to have a good time optimizing the stage.