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When Balon Greyjoy, Lord Reaper of Pyke, first rebelled against King Robert Baratheon and declared himself King of the Iron Islands, his brothers Euron and Victarion Greyjoy carried out an attack against the Lannister fleet and burned it at port, in order to prevent it being used against the Iron Islands. This was, however, the only ironborn's victory during the rebellion; Stannis Baratheon led the Royal Fleet to victory against the ironborn.
In the books
In the A Song of Ice and Fire novels, no information is given about the Lannister fleet (like the number of its ships), except that it was destroyed by Victarion Greyjoy during the Greyjoy Rebellion.
According to George Martin, the Lannister fleet consisted of 20 - 30 ships, and Tywin replaced the ships that were destroyed by the ironborn. Such a small fleet could not be used for fighting a major battle; in such case, the Lannisters would have to depend on the ships of their bannermen.