I agree with Sarah that my understanding of Buddhism just doesn’t work without accepting rebirth and karma. Those two go together and don’t work on their own. I have read works on so-called Western Buddhism by such authors as Stephen Batchelor and Robert Wright, but to me that is just therapy, which is valuable in it’s own right of course.
I’m not sure I like your phrase ‘manage to convince yourself’. I think the very first thing is to acknowledge that there is no evidence against rebirth and no particular evidence for the view that the continuum of consciousness completely and utterly ends with death. As a person raised in the materialist/atheist UK tradition, this was the first and most difficult step for me, as it seemed overwhelmingly like a commonsense position against which everything else was just superstition and wishful thinking.
Once I had come to the conclusion that this materialist view of consciousness was just a theory like any other, I was able to free myself from what I came to see was just a learned bias. After that, my confidence in the dharma teachings and in the teachers, who always seem so beneficial and so wise, gradually drew me in to the position where I think it is more likely than not that our most subtle consciousness does carry on after the body dies, and that what we have made of ourselves in this life has a fruition in the next.
Of course, it would be different for someone brought up in a religious tradition and I have no advice for that.