Most translation contain the word "But" at the beginning of the verse.
Is it present in the manuscripts that are available to us?
Fewer translations contain "And" instead.
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It's the waw translated "but." The most common translation for the prefixed waw in the Tanakh (Old Testament) is "and." Here in Jonah 1:3 the Jewish Publication Society translation translates it "however." See the quote from BDB below.
וְ is used very freely and widely in Heb., but also with much delicacy, to express relations and shades of meaning which Western languages would usually indicate by distinct particles. But in Heb. particles such as אוֹ, אָז, אַךְ, אָכֵן, אוּלָם, בַּעֲבוּר, לְמַעַן, לָכֵן, etc., were reserved for cases in which special emph. or distinctness was desired: their frequent use was felt instinctively to be inconsistent with the lightness and grace of movement which the Hebrew ear loved; and thus in AV, RV, words like or, then, but, notwithstanding, howbeit, so, thus, therefore, that, constantly appear, where the Heb. has simply וְ. -- Brown, F., Driver, S. R., & Briggs, C. A. (1977). Enhanced Brown-Driver-Briggs Hebrew and English Lexicon (p. 252). Oxford: Clarendon Press.
Jonah, however, started out to flee to Tarshish from the LORD’s service. He went down to Joppa and found a ship going to Tarshish. He paid the fare and went aboard to sail with the others to Tarshish, away from the service of the LORD. -- Jewish Publication Society. (1985). Tanakh: The Holy Scriptures (Jonah 1:3). Philadelphia: Jewish Publication Society.