Depending on the content, you could pay for the proper licenses needed.
If the copyrighted material is in ContentID, oftentimes the owner won't issue a copyright strike, but instead do a 3rd party claim and monetize the video on their behalf. That way they receive any income generated by the video instead of the channel who uploaded it.
More than likely, this is what has happened to the content you've seen. Otherwise, it's just a matter of time before they're found and dealt with accordingly.
I did accidentally once, I didn't know too much about anything lol I know a little more now. I didn't get a strike they just said I wouldn't bet any money for the video if I had ads on it which I don't
Ubless you're super rich and can buy the license to 'hit me baby one more time' then I don't think there's a free way as it were. to be honest I use a handful of brilliant artists on soundcloud that I credit in my description. I wouldn't even entertain using copyrighted material without written consent