Song purchase is a license for personal use, not for public broadcasting. Now, in the case of many DJ's, there is often a case under fair use that they use due to remixes utilizing very small portions of other content mixed with largely generic or customized beats and backing tracks. To use a silly example, if you wrote your own industrial mix and overlaid something like the line "Can't touch this" by MC Hammer as a part of a much larger creation of content that you made yourself, it is unlikely that MC Hammer would have a legitimate case to sue you for infringement. Fair use is however only a legal defense in a court of law and does not apply to private platforms.
YouTube adheres to fair use only in a very limited sense. ContentID is the system which matches content to known sources and it does not discriminate on the basis of perceived fair use. You could have a 15 minute long song with just a 5 second clip of some other material and if YouTube recognizes it, your entire song will be claimed by the owner of those 5 seconds.