When the British took over the area from the Ottoman Empire, they did indeed begin to allow Jews to immigrate from Europe and elsewhere into the area. They considered it important: the British had committed themselves to helping create a "Jewish national home" in the Balfour Declaration, and the League of Nations explicitly endorsed this. The text of the British Mandate, which included provisions related to management of this Mandate, included provisions such as:

Whereas the Principal Allied Powers have also agreed that the Mandatory should be responsible for putting into effect the declaration originally made on November 2nd, 1917, by the Government of His Britannic Majesty, and adopted by the said Powers, in favor of the establishment in Palestine of a national home for the Jewish people, it being clearly understood that nothing should be done which might prejudice the civil and religious rights of existing non-Jewish communities in Palestine, or the rights and political status enjoyed by Jews in any other country

And also:

The Mandatory shall be responsible for placing the country under such political, administrative and economic conditions as will secure the establishment of the Jewish national home, as laid down in the preamble, and the development of self-governing institutions, and also for safeguarding the civil and religious rights of all the inhabitants of Palestine, irrespective of race and religion.

This was also echoed in Article 6. The British were, in that part, to coordinate Jewish immigration:

The Administration of Palestine, while ensuring that the rights and position of other sections of the population are not prejudiced, shall facilitate Jewish immigration under suitable conditions and shall encourage, in co-operation with the Jewish agency referred to in Article 4, close settlement by Jews on the land, including State lands and waste lands not required for public purposes.

However, over time, the rise of Jewish immigrants led to backlash. Among the grievances, besides the sense that Jews getting a state there would be at the expense of Arabs, also included some level of economic backlash. Jewish immigrants frequently purchased land from Arab landowners who did not live in the area, and employed local Arabs to farm the land and live on it. Jews purchasing the land would often want to work the land themselves, in line with socialist ideals, and thus would fire and remove the Arabs who lived on the land they bought. This led to local Arab farmer backlash, since while they did not own the land, they had often lived on it for quite some time, and felt the displacement unjust.

The reason I mention this policy is because the backlash led to multiple outbreaks of violence against Jews, and against the British authorities. The major outbreaks include riots in 1920, 1921, and 1929, with 1929 involving the particularly notable massacre of Jews in Hebron. In 1936, these tensions spilled over into outright revolt against the British. While the revolt was largely over besides roving militia bands soon, it continued formally until 1939.

At that point, given the rising tensions with Germany, the British were eager to placate the Arab population and avoid a revolt that would help the Nazis and divert British resources. Thus, the British released a white paper that limited Jewish immigration in 1939 for 5 years, i.e. during the midst of the Holocaust, to a total of no more than 75,000. That same white paper did indeed also limit land purchases by Jews to certain areas and amounts.

These policies stymied some level of immigration, though many still arrived illegally, particularly fleeing the Nazis. They also stymied land sales. Despite Jews owning only 5-6 percent of the land by 1947, the backlash was indeed severe enough to result in this British policy by 1939.

Thus British policy was not stable, or the same, throughout the 25+ years of administration, so I've done my best to give you the broad strokes of it, and can answer more questions if you have any.