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Written Aramaic translations of the Hebrew scriptures (which is what "targum" generally refers to today) did not yet exist at the time of Jesus.
From Claude Tresmontant:
There were oral translations in Aramaic of the sacred books written in Hebrew; they were called targumin. A translator in the synagogue would read aloud, translating a passage from the Torah or one of the prophets. But in the era before the destruction of the Temple, putting these translations into writing was formally prohibited." (The Hebrew Christ p. 5)
So the scrolls in the local synagogues would generally be documents written in Hebrew, from which Aramaic oral translations were sometimes given.
Near the time of the destruction of the temple (AD 70), written Aramaic translations started to be used. During the early first century, the Greek Septuagint was the only accepted, authoritative, written alternative to the Hebrew Tanakh for Jews in/around the Roman Empire.
Oral translations of the Hebrew scriptures into Aramaic came much earlier, as the Aramaic language spread throughout Middle-Eastern Judaism following the Babylonian conquest in the 6th century BC.