Where is the Lannister fleet?

Question

The Lannisters have historically maintained strength on the seas, which is crucial for defending their coastline. Especially major trading hubs like Lannisport, and to defend the Rock herself, which cannot be besieged without a naval blockade.

We see the Lannister fleet in action during the Pre-Conquest Lannister invasion of the Iron Islands and the Greyjoy attack on the Lannister Navy during the Greyjoy rebellion. After that there is no mention of the Lannister fleet at all.

Later when Tyrion and Cersei devise their plans for the War of the Five Kings and the invasion of Dragonstone, neither of them considered the naval power of the West herself but the former considered using the Greyjoy navy and the latter relied upon the naval power of House Tyrell and her banners.

So where is the Lannister fleet? Is it too weak to be used in an offensive role, if it exists at all? Or did Tywin just forget to rebuild their power at sea, which is fairly unlikely.

Answer 1

George R.R. Martin shed some light on the Lannister fleet in correspondence with a fan.

The fan asked:

Q: Mr. Martin, we found it peculiar that Tyrion considered the importance of the Greyjoy fleet in aCoK (when Balon's message reached him in KL), since he never thinks about the West's own naval strength and it's usefulness against the Starks.

So we began to doubt said strength.

Did Tywin ever rebuild the Lannister fleet, after Euron burned it eight years ago?

And if he rebuilt it, did he ever appoint a permanent commander of this fleet?

GRRM replied:

Yes, Lord Tywin certainly replaced the ships that were lost. [...]

The lords whose lands abut the coast of the Sunset Sea all keep a war galley or three about for coastal defense, and of course those shores are home to scads of fishing boats as well. The Lannisters have a larger and much grander fleet, but we're still only talking about twenty to thirty ships, perhaps. To fight a major battle, they would call the ships of their various bannermen, just as Stannis summoned the lords of the narrow sea for the battle on the Blackwater.

For what it's worth, however, their ships would be larger and more formidable than the longships of the ironmen -- cogs, carracks, and war galleys of various sides, up to the great dromonds with scorpions and catapults on deck.
_{The Citadel: So Spake Martin - Lannister Fleet}

So to summarise:

1. Tywin Lannister indeed rebuilt the Lannister naval power to some extent.
2. The Lannister fleet numbered somewhere between 20 to 30 although those 20-30 ships would be much bigger and much more formidable than the Greyjoy Iron fleet.
3. Local lords with coastal holdings would maintain 1 to 3 War galleys (again bigger and more formidable than the smaller Greyjoy longships), which can be summoned by the Lord of the Rock. So assuming that the Lords of Crakehall, Crag, Fair Isle, Kayce, Banefort and Frostfires maintain some galleys, the total sum of Lannister naval vessels would be somewhere between 26-38 or 36-48. Lannisport is also a major trading hub so a number of ships could be commandeered from the merchants in case of necessity.

As for why have the Lannister hadn't used their own fleet, I believe the following would be the reasons:

1. Deploying the Lannister fleet elsewhere would leave Casterly Rock, Fair Isle, Lannisport etc vulnerable to invasion and raids from the Greyjoys.
2. Before the alliance with the Tyrells, the Lannister fleet might have had to fight the Tyrell fleet in order to pass through to the East Coast if they were seen by Tyrell watchtowers on the Shield Isles and elsewhere.
3. It would have been too risky and too expensive to move the Lannister fleet from the West coast to the East coast given the risk of battles and violent seas.
4. With the majority of the Royal fleet defecting to Stannis Baratheon, the Lannister fleet would have been outnumbered and possibly wasted in battle.
5. After the Tyrell alliance, Lannisters did not need to redeploy their own fleet to the East coast and relied on their allies' naval power instead.

Answer 2

According to your second link the Lannister “fleet” that was destroyed by the Greyjoys was twenty or thirty ships. That’s not much of a fleet by Westerosi standards —- Stannis Baratheon’s fleet in the Battle of the Blackwater had two hundred ships (A Clash Of Kings chapter 58). Similarly the Redwyne Fleet, sworn to House Tyrell, also contained two hundred warships. So even if the Lannister fleet was entirely rebuilt after the Greyjoy Rebellion, it was simply too small to have much military significance.