First off moving nine million people anywhere in a short period of time (at the rate of several millions a year) creates a mess no matter who and no matter where (look at the partition of India and the movements of Hindus/Muslims across the borders). As another poster has mentioned, it wasn't until after Kristalnacht in Germany, and the Anschluß in Austria that the German and Austrian Jews were in a mood to emigrate. The Jews of Western Europe (Norway, Denmark, Holland, Belgium, France) were not attempting to leave (for anywhere), why should they none of them were expecting the Nazis to conquer their countries. In the 30s even the Jews of Italy were in no mood to leave, and they could without confiscation the significant Italian anti-semitic measures were later on at German urging. The picture in Central and Eastern Europe was more mixed, places like Czechoslovakia were not "unpleasant" for Jews, Poland and Romania were more likely to see Jews with means leave if they had a place to go, others were mixed. Of course the USSR you could not leave.
Prior to the outbreak of war, the only Jews with a significant push to emigrate would be those of Germany and Austria, and then Czechoslovakia after the full occupation. Once the war starts, the regions with the most Jews, Poland and the Western USSR, the Jews can't leave. Likewise the Jews of Western Europe after spring 1940, and most of those Jews won't flee between September, 1939 and June, 1940. After all Norway, Denmark and Holland are neutral (and sat out WWI) and French and Belgian Jews will defend their countries. With Italy at war with the Allies, those Jews are trapped - maybe they could go to the USA or Argentina, maybe. The Jews of Yugoslavia, Greece, and Bulgaria are potential immigrants. Hungary and Romania, very iffy because of close ties to Germany, although until BARBAROSSA letting Jews out to go to British territory is not totally ASB.
If you look at the numbers, even had the UK completely dropped any restrictions on Jewish immigration to Palestine in the post WWI period, following through on the Balfour Declaration, nowhere near nine million Jews would come in to Palestine between 1920 and 1939. Perhaps 400,000+ came between 1920 and 1945 (both legally and illegally). If ALL the Jews in Germany/Austria/Czechoslovakia went to Palestine in the 30s before WWII started in 1939 that would be around 1.1-1.3 million Jews, if ALL the Jews in Italy, Yugoslavia, Greece, and Bulgaria did likewise that might make another 750,000 or so. What this means is under any sort of realistic open immigration policy, and throwing in some Jews from everywhere else except the USSR, perhaps two million Jews might have immigrated to Palestine in the 30s over and above what happened - all of the Jews in those named countries won't leave, and some of the Jews in other countries (less the USSR) will come.
Using funds/means brought by these Jews as well as foreign Jewish support, integrating 2 million Jews 1930-1940 will be difficult, a lot of strains but doable. After the war starts, and after the fall of France, an open policy will allow some Jews in - French Jews who escape to Vichy territory may find it possible to get to Palestine. The reality is by the time The Nazis bag the Jews of Western Europe, Poland, and the Western USSR it is too late. While the governments of Hungary and Romania might be happy to see the Jews leave, going to a British territory is not going to happen, and after 12/41 the USA (even if it would take them which it would not) is also out of bounds. Sadly this means perhaps four million Jews die in the Holocaust, two million are saved, but really no more.
The reality is that many Jews did not have the means to emigrate and/or the will to do so. The Holocaust was not expected,the early Nazi depredations were just more of the same old stuff brought back, and in any case that was for the Jews of Germany (and later Austria). Only if the Jews in areas that were occupied/cooperating during WWII were told about the Holocaust and believed it would the bulk of the Jewish population of Europe pick up and leave, and not all would go to Palestine if they had a choice.