Why does Ash keep getting new friends and leaving his old friends?

The out-of-universe reason is that the anime writers need to be able to to showcase the characters from whatever generation has just been released. The anime is, after all, just a 1000-episode-long ad campaign for the games. Ash has had 16 friends as of the end of Sun & Moon; if they were all still travelling with him, the cast would be almost unmanageably large.

The in-universe reason is that, almost without exception, Ash's friends choose not to follow him when he travels to a new region, instead preferring to go off and do their own thing. Misty took over the Cerulean Gym, May and Iris went to Johto, Serena went to Hoenn, and so on. You could spin this as Ash's friends leaving him, not the other way around. Either way, he certainly doesn't forget his old friends, as several of them, especially Misty and Brock, have shown up again after their initial departures from the group.