About whether you have the same thing as most of us on this forum have, I want to add that most people have the problem for a long period, like months or years, so it may be too early to say whether you have the same thing. But if rare lesions exactly match with other rare lesions (from a visual perspective), I would expect it to be the same kind of problem.
Benzoyl peroxide, what your doctor advised, may help for some people, as I have seen a few positive reports about it. But it may dry your skin, which could make it worse from that perspective. Maybe combining it with a moisturizer helps?
Oral antibiotics may provide temporary relief, as seen in the reports on this forum.
If it’s getting worse on the treatment of the doctor, I would suggest to go to him/her again even before the recommended end of a month.
Note that there exists an form of Soolantra (topical ivermectin) meant for animals. It’s available without prescription and in liquid form. For more info, see: Does yours spread?
It’s difficult for me to advice against things a doctor recommends, as I’m not a doctor myself, and have far less knowledge about medicine than doctors. So I can only tell my experience and about the experiences that are common on this forum.
For the skin problem this forum is about, it’s important to be gentle for the skin. In my case, anything harsh (even a harsh or dry environment) can cause outbreaks and worsenings (when I had the problem, I’m currently lesion free for about 1.5 year). Hibiscrub (which I also used myself) is a bit harsh, it can easily dry out the skin, making it worse.
This is a bit of a paradox: keeping the skin as clean as possible (with things such as Hibiscrub) removes/reduces bacteria, dirt etc, but on the other hand it dries the skin which makes it dry, thinner and itchy and thus less protected against bacteria. On the other hand there is moisturizing the skin, which makes it easier for the skin to heal itself, but this may also make it easier for bacteria to survive and for dirt to stick.
In my case, moisturizing is better than cleaning the skin. For oozing lesions on the scalp, I would go for a mild treatment: washing it with lukewarm water and a gentle mild shampoo. Using a shampoo every day, could be too harsh.
Also try to avoid scratching lesions/skin.