The "Late Palaeolithic Extinctions"
The theory of "overkill" by human hunters as a likely explanation for the extinctions was suggested by Paul Martin as long ago as 1967. It is unfortunately still not generally accepted and writers on evolution tend to skirt round the subject with great caution. See Wikipedia articles.
The obstacle to its acceptance is perhaps a failure to recognise the effectiveness of human weaponry and co-operative hunting techniques during this period. What happened in America happened only about 10,000 years ago. The invaders of America and their Eurasian contemporaries were not primitive - they are likely to have been organized in large tribal groups and they certainly had relatively sophisticated weapons. The settlement at Sungir, where a large community hunted mammoths and organized the production of ivory beads on a large scale had been established thousands of years before the hunters invaded America.
One problem remains - how did the elephants and many other large animals survive in Africa and (to a lesser extent) in southern Asia? In certain places (e.g. India) it is possible that deliberate conservation (for hunting by ruling elites) was a factor.