I think the issue here is the timeline. Hitler came to power in Germany in the early 1930s, invaded Poland (which had a large Jewish population, much larger than that of Germany) in 1939, and invaded the USSR (which had a similarly large Jewish community) in 1941. Intense persecution (ghettoization, etc) and mass killings began quite quickly after Germany occupied Eastern Europe, particularly after the invasion of the USSR. So Jews simply did not have very long to escape. Israel declared independence in 1948.

At the time Israel was controlled by the British under the Mandate for Palestine, which outlined the British role as running the place and transitioning from Ottoman rule to independence. But what that independence would look like was already a strong matter of contention, with conflict between the Arab and Jewish populations growing, with riots in beginning in 1921 and increasing levels of violence following that with a major Arab uprising in the 1930s.

The British wanted very badly to stabilize the situation, but were unable to come up with a long-term solution amenable to the population. Their attempted solution was basically to maintain the status quo until a solution could be found. The primary way they did this was by issuing successive White Papers, which restricted Jewish immigration to Palestine.

While there was a wave of Jewish immigration from Germany in the 1930s, its size was limited by British regulations. And from Eastern Europe there simply wasn't time, and the British rules made that impossible anyway.

The British immigration restrictions were extremely unpopular among Jews, who fully appreciated that these restrictions kept Jews from fleeing the Nazi's rise (though the extent of what that meant was not fully understood yet). The Jewish establishment in Palestine mounted a significant campaign of resistance, the centerpiece of which (at least in the late 1930s and early 1940s) meant a massive system of illegal immigration. But with the enormous number of Jews under Nazi rule in Eastern Europe it simply wasn't practical to evacuate a significant proportion of people.

tl;dr Hitler's rise triggered a wave of immigration of German Jews in the 1930s, but the British restricted immigration at this time. Also, there were simply too many Jews in Eastern Europe for a significant proportion of them to escape by migrating.

There were a few major reasons:

First, there were no permits. In 1939, the British released a white paper policy (usually referred to as the White Paper of 1939) which, among other things, included a provision stating that only 75,000 permits would be given to allow Jewish immigrants into Palestine. Fewer than 16,000 illegal Jewish immigrants came to Palestine between 1939 and 1945, when there were 40,000 between 1934-1938. The noose was tightened.

But even this doesn't fully explain the issue, because the 75,000 permits were not all used. It is true, however, that the British began subtracting illegal immigrants from the total permits allowed (so, for every illegal immigrant a permit would be removed), which may account for at least part of this. But the British were, without a doubt, trying their best to stop refugees from reaching Palestine. They said they didn't mind immigrants leaving, but not to Palestine, and this led to the Irgun (a small Zionist group) undertaking violent action that frequently resulted in British deaths, though it halted this for some years during the war as well in preference of not harming the war effort (and the main Zionist group, Haganah, went on to help the British arrest its members nearer the end of the war).

In the meantime, the British did their best to pressure Turkey and Romania and the like to stop the refugee ships from leaving for Palestine in the first place. They also briefly tried to deport the immigrants to their countries of origin, but this failed miserably and got only about 100 immigrants deported home over two years. Instead, thousands were shipped by the British to camps in Cyprus and Mauritius, and it showed. By March 1944, only 55,000 legal and illegal immigrants of the 75,000 limit had arrived in Palestine, but fighting the British was not popular among the Zionists. The mainstream Zionist leader who would become Israel's first Prime Minister, David Ben-Gurion, said:

We shall fight the White Paper as if there is no war, and we shall fight the war as if there is no White Paper.

But the fight was restricted to immigration (the White Paper also contained provisions that restricted where Jews could buy land, for example) and was largely left aside as the war effort took precedence.

Jews did flee, however, if not to Palestine, in the years before the war. There was a sharp jump in immigration between 1933 and 1935, with over 140,000 Jews reaching Palestine during those years. Another 32,000 entered in 1939, though the years between 1935-1939 were a little slower in total immigration. But as can be seen, from 1934-1938 there were over 100,000 Jewish immigrants, of which 40,000 were illegal. But when legal immigration was curtailed, and illegal immigration got the squeeze, Jews were left out of Palestine.

Why they were still in Europe is another question, but that's why not Palestine at least.