An inconvenient truth in Buddhism, the Buddha did not reject reincarnation/rebirth. Despite this, Western Buddhists want evidence for rebirth. In other words, they want an empirical case to be made. So what would be an empirical case?
Believe it or not we have one such case. It is to be found in the work of the late psychiatrist Ian Stevenson of the University of Virginia who studied cases of children which strongly suggested reincarnation (one famous book of his is Children Who Remember Past Lives). While he has never claimed to have proven reincarnation beyond a shadow of a doubt, he does show that it is reasonable to believe in it. This is something I would guess that probably irritates rebirth deniers who count themselves to be the holy friars of scientism.
Another irritation, is the book Death and Personal Survival: The Evidence for Life After Death by Robert Almeder that, incidentally, Buddhist rebirth deniers most likely know little or nothing about. Professor Almeder is well qualified to judge just what is credible in making an empirical case for reincarnation. He has laid out eight conditions any one of which strongly suggests reincarnation. One condition is the supposed reincarnate has to demonstrate a complicated skill such as a foreign language that could not have been acquired in this life.
By no means are the eight conditions conclusive proof. By the same token, fulfilling some of the conditions, reincarnation is not improbable which means it is reasonable to believe in reincarnation from an empirical standpoint.