It is not very old, because there is a story about it ... in the 19th century a certain Karmni Grima had a vision and decided to build a shrine...
And from that it became a big nice church were a lot of Catholics go to pray our Lady which is miraculous. I'll try to find some more accurate info,
construction of a magnificent church in Romanesque Style began in 1920. In 1932 the new Church was blessed and opened to the public and in 1935, Pope Pius XI raised it to the dignity of Minor Basilica. In the same year the miraculous image of Our Lady was crowned by Cardinal Alessio Lepicier, by decree signed by the Secretary of State Cardinal Eugene Pacelli later Pope Pius XII.
In May 1990, Pope John Paul II came to Ta' Pinu Shrine and after praying in the Chapel he celebrated mass in the forecourt and decorated the image of Our Lady with five golden stars, as testimony of our people's love to the blessed Virgin Mary.
While on 18 April 2010 Pope Benedict XVI, on his first visit of Malta, presented a golden roses. While addressing the crowd gathered at Floriana, before he recited the Regina Caeli Pope Benedict expressed his pleasure: "I am aware of the particular devotion of the Maltese people to the Mother of God, expressed with great fervour to Our Lady of Ta' Pinu and so I am pleased to have the opportunity to pray before her image, brought here specially from Gozo for this occasion. I am also delighted to present a Golden Rose to her, as a sign of our shared filial affection for the Mother of God. I ask you in particular to pray to her under the title Queen of the Family..." [source]