Since the 1950s it has been widely accepted within the Iraqi Jewish community that migrated to Israel that at least some of the five bombings of 1950-51 were linked to the Zionist underground active in Iraq. The Iraqi government itself put on trial and hanged those they claimed to be responsible, while the Israeli government launched a couple of investigations into the matter in the 50s and 60s, determining that they bore no responsibility. Various historians throughout the years have concluded that Zionist involvement was likely in some of the bombings (e.g. as early as 1970 in an essay titled "Minorities" the conservative Iraqi-Jewish historian Elie Kedourie determined that the Zionists were "certainly capable" of such actions).
Avi Shalim claims to have found conclusive evidence confirming Zionist involvement in three of five bombings that took place during 1950-51. I have read his recent memoirs where he makes this claim, and although I'm not qualified to reliably judge on the credibility of the evidence he presents, I do find it compelling (it's based on testimony given to him by an elderly Israeli man who was involved in the Zionist underground in Iraq, and who, Shlaim claims, presented him with an Iraqi police report produced at the time confirming that the Zionists had compromised a member of the Baghdad police force and bribed him to help execute the most destructive bombing that killed four Jews at a Synagogue in January 1951).
However, I don't accept Shlaim's claim that the bombings were the primary cause of the flight of Iraqi Jews to Israel (and as Shlaim himself acknowledges, he is not an expert on Iraqi history). The first bombing took place in April 1950, just after the passing of a law allowing Jews to denationalize and move to Israel (low-scale but growing illegal emigration via Iran had been underway since 1948). Shlaim concludes that this particular bombing was the work of members of the Arab nationalist Istiqlal Party. Between that time and January 1951 when the second bombing at the synagogue happened, around 86,000 Jews - the vast majority of the community - had registered to leave, and were waiting in desperate conditions in Baghdad for their journey to Israel to be facilitated. Shlaim's own family left in the summer of 1950 (he was four years old). So by the time the bombings that can be credibly linked to the Zionists happened, the exodus of Iraqi Jews was already near completion. In any case, a few minor bombings simply cannot explain the decision of well over 100,000 people to pack up and flee the country they'd called home for generations.
The question then arises - what was the purpose of the bombings of 1951? It seems unlikely that their purpose was to drive the Jews out of Iraq since (a) that was already happening, and (b) the Israeli government was facing major problems at that very moment receiving, processing, and housing hundreds of thousands of other migrants from Eastern Europe (especially Romania and Poland). Shlaim acknowledges that there is no evidence to suggest that the Israeli government had a direct hand in the bombings. Other historians have speculated that the bombings were an attempt to draw the attention of the Israeli government to the plight of the Iraqi Jews then in limbo in Baghdad and hurry up their departure, as the Israelis had prioritized the incorporation of the aforementioned European migrants ahead of them. But until further evidence comes to light, it's impossible to say for sure.
Ultimately, the flight of Iraqi Jews was driven by a growth in fear and uncertainty about their future in Iraq that had developed among them over the course of the previous decade and a half...
The broad context was the growth of Arab nationalism in Iraq and the brewing conflict with Zionism in Palestine. There were lingering resentments at the Iraqi Jews' pro-British stance during the mandate years (the Assyrians had paid for their own pro-British stance with a horrible massacre in 1933). And increasing suspicions that they were in bed with Zionism further fueled a general growth in hostility directed at them, with very little distinction made between Jew and Zionist by hostile nationalists (in fact, the Iraqi Jews had little interest in Zionism at all, at least until the Farhud, and were culturally embedded in the Islamo-Arab society of post-Ottoman Iraq). Measures directed against them in the mid-30s included the banning of the teaching of Jewish history and Hebrew in Jewish schools, while Iraqi nationalist media increasingly agitated against them, especially after the outbreak of the revolt in Palestine in 1936. There was some mob violence aimed at them, and occasional murders, although the level of violence depended on which particular government happened to be in charge.
The Farhud was the culmination of this general growth in hostility, but it didn't prompt the immediate flight of the community. However, it provided a convenient episode which Zionist emissaries could point to in order to try convince the Iraqi Jews that they had no future in Iraq, and from 1941 onwards there was a small growth in Zionist activism among young Iraqi Jews.
Things came to a head with the 1948 Palestine War, which the Iraqi government used as a pretext to directly target the Jews, having come to determine that the national interest would be served by the departure of a community increasingly understood as a potential fifth column, either due to the perceived threat posed by Zionism or by the disproportionate Jewish involvement in the Iraqi Communist Party (one of the biggest Communist Parties in the Arab world). Prime Minister Nuri al-Said even planned a population exchange of Iraqi Jews for Palestinian Arab refugees, and it was his measures to freeze and confiscate the assets of the departing Jews in 1951 which brought the saga to a close. By 1952 the Iraqi Jewish community had effectively ceased to exist.