You are uninformed. Eastern European Jews had been migrating to Ottoman Syria in significant numbers since the 18th century, and in greater numbers at the turn of the 20th century. Due to increasingly violent pogroms and hostility in Russia, mainly (but not only). The amount of them that went to Palestine was small compared for instance to the US because, simply, Palestine was a poor and unimportant backwater province of a decadent empire. It's because they were organized that Western powers decided to help them, not the other way around. And perhaps, maybe, who knows! what happened in Central and Eastern Europe during the 40's also was an incentive to go somewhere where they could claim to be the majority, for the first time in centuries...