In the 9th century, a Jewish Rabbi forms a new branch of Judaism - which is more open towards converting people to Judaism and which changes a few other things, but is still definitely Jewish. At that time, there was a lot of tolerance towards other religions by the Muslim authorities, even pretty equal access to political institutions, but conversion will probably still need to be done in secrecy as much as possible. After a few years, once the following is large enough, there is also an active policy of convincing Jews from areas with lots of intolerance to come to the place.
The Religion grows quickly, because of the greater tolerance and all the opportunities in the "golden age" beginning there at the time.
In the 10th century, a member of this religion becomes General of an army of the Iberian Caliphate. He successfully fights a few wars or revolts and is rewarded with the rule over a small province in the North of Spain, bordering one or two of the Christian kingdoms there.
He invites Jews from all over Spain and other places to live there and sees for the conversion of many of the locals (Christians and Muslims) to Judaism. The local capital becomes a prospering city. To feed all the immigrants, he invests a lot of effort into building irrigations, dams, markets, streets, and so on. A province of a few hundred thousand people quickly reaches the first million.
At the same time, this brand of Judaism spreads to many other places and gradually becomes the dominant form of Judaism in Europe, North Africa, and the Mideast. There is also some exchange with the then-still-existing Khazaria. To avoid unnecessary wars with Muslims and Christians, conversion of people from those religions is kept at a low flame - as much as is tolerated in the different places. Also, he arranges for the movement of Jews from intolerant to more tolerant places (and his province), thereby partly helping defusing tensions between Christians and Jews as well as Muslims and Jews.
Mountains in some parts of the province provide a good protection agaist invaders, while the large number of people allows a big army, which is hardened for battle by doing a lot of the infrastructure work in peace times.
After a few years, a battle with one of the Christian kingdoms erupts, and, with the help of Muslim Spain, the Jewish province wins and the Kingdom falls. To avoid clashes with Christians, the ruler of the Jewish state, which now is a kingdom, sees for the state to be split into pretty independent Christian and Jewish provinces, expanding the Jewish area only slightly. There are also one or two provinces which are supposed to be liberal heavens for Jews, Christians and Muslims to live together. And the Jewish king lets the Muslims have most of the conquered territory in exchange for complete liberalism in provinces surrounding the Jewish kingdom, so that the expansion of the Jewish territory doesn't get that much attention.
At the same time, measures are taken to neutralize Christian and Muslim (religious) leaders who create tension between Jews and their religions with hate speaches, suprressive acts, and the likes - war mongerers are involved in scandals, their (mainly non-Jewish) opponents miraculously get to know a lot of their plans, authors and other opinion makers of the different religions who advocate peace and a liberal attitude among the different religions get some support, and so on.
In the next war, the remaining Christian Kingdoms of Spain fall. Directly after the war, a quarrel erupts over how to share the conquered territories. The quarrel turns into a war, which the freshly mobilized Christian forces from France join. In the later peace, one of the Christian kingdoms gets her independence back, the Jewish Kingdom expands a lot, the French (actually Franks) get some territory, Muslim Spain is reduced to the remains.
Pretty soon, the next wars follow similar to OTL, but as Judeo-Christian alliances. In those wars, the Franks capture what was Aragon at her greatest extent, the Christian kingdom in the north stays the same, and the remains of the Iberian peninsula go to the Jewish kingdom.
Because of the tolerance of the Jews against the Spanish Muslims, relations to the Muslim world recover quickly. So when later a war against the Christians break out, they send a lot of help, also because of their interests in the Middle East. Therefore, the Jewish Kingdom takes the remaining Spanish Christian kingdom, Aragon and even some territory beyond the Pyrenean Mountains. In the later peace, the Pyreneans are declared the border between the Franks and the "Juberians".
The plague weakens all sides considerably.
Juberia manages to get into the same great power status as Spain IOTL, but with added Portuguese ressources. Because of the religious isolation and growing power of Europe, there is not much interest in engaging in European affairs, therefore no Spanish Armada or the likes. Juberia becomes a sea faring nation with lots of trade all over the Mediterranean, northern Europe and western Africa. Colonies spring up in western and later southern Africa similar to OTL. Jewish explorers discover America. Because of the scarcity of labor after the plague, African and Amerindian slaves are imported. After a few years, demand for people in America grows, so that African slaves go mainly there. After the population has reached pre-plague levels again, a lot of Juberians go to America, too, to escape poverty at home. Criminals are also brought there, to increase colony sizes for more tax revenue.
Later, people from all over Europe, Africa and Asia are invited to come to America. A melting pot policy makes sure America stays under Juberian control. Other European powers settle mainly in North America, similar to OTL.
Juberian naval explorers cartograph the whole world and discover Australia, decades before the British (OTL, a Portuguese ship wrecked there in the 16th century - so it's not unlikely). Australia is settled by Juberians.
The decline of the Muslim world makes eastern Africa, the Mideast, the Balcans, south-west Asia, and north Asia parts of the Christian sphere of influence, while north-west Africa is controlled by Juberia.
As Juberia is quite a lot stronger than Spain and Portugal together, some of their historical defeats are avoided - no British Gibraltar, many more parts of Africa become and stay Juberian, Juberia develops in a similar fashion to Britain.
In the 18th and 19th century, the different independence movements take their toll - Juberian possessions in Amerika form independent states, only the African colonies and Australia stay in Juberian hands for now. Britain faces similar problems. But friendly relations with all the Jewish states and colonies all over the world make sure Juberia is pretty save from any attack.
During the 20th century, African and Australian nations get independent, too, maybe caused by a communist SU and republican US and South Americas, similar to OTL. If Nazi-Germany is also not butterflied away, most Jews find a new home in one of the many Jewish countries all over the world, the anti-Jewish propaganda leads to diplomatic clashes with the Jewish countries and their allies, and war is declared as soon as the extinction efforts against the remaining Jews become known (probably during operation Barbarossa or sth. similar, 1941). Therefore, wwii ends much sooner than OTL.
Today, there are about as many Jews on the planet as Christians or Muslims - in this TL.