No, no, no, no, NO!! A 'Roman galley' is NOT anything like a Byzantine galley, nor is a Carthagenian, or Greek galley anything like a Byzantine galley! By the time of the Middle Ages, the galley had been significantly altered in both form and construction!!! The closest comparison for a Byzantine, or post-classic Mediterranean galley with anything you might find in the hobby shop is the Heller model 'La Reale,' which although it is of the 18th century, it is closely based on a Mediteranean galley type which had been around for almost 500 years! The Spaniards, French and Maltese continued to produce and use these galleys, partly because the Turks and Barbary Corsairs continued to do so (and had done since at LEAST the sack of Constantinople/Istanbul, and partly because until the age of steam, nothing was really as effective for use in shallow waters and fickle winds as a galley in the Med (close design copies were also used in the Baltic, and for the same reasons).
Finally, there seems to be a LOT of assumptions flying around as to what a Viking ship, or could not do, and/or their relationship to the Normans. The Vikings were QUITE capable of transporting horses in their ships (they brought them all the way to Iceland and Greenland, thank you very much!), along with cattle and other livestock. It should be understood that the Vikings had different ships for different purposes, though they were all built with the same techniques. The Drakkar or Long Ship was primarily a warship, and these ships could get very large indeed. Although the academics for years have debunked the various claims of ship sizes in the ancient Sagas, in fact, Archeologists have just excavated a Drakkar in Ireland that was over 180' long! That's plenty of space to stow horses, and pretty much anything else you might want to bring along. The Drakkar is in fact very much like the Oseberg ship in configuration, very long, lean, and very fast. In fact, the 'Viking ship' has never really gone out of fashion in Scandinavia, and the most recent version, known as a Nordlands boat have recently had something of a revival, a number of which are members of a club in Norway, that race quite regularly.
Oh and by the way, when I say these Viking craft, both modern and 'ancient' are fast, I mean they can sail at up to 20 knots!! For many years, I often wondered how it was that the Vikings would be able to surprise so many towns and monasteries on the coast with essentially, glorified rowboats. You would think it would take HOURS for these things to get to the beach, which should allow plenty of time for the local warriors to assemble to drive them back into the sea. But if you start thinking of the Drakkar type, not as a lumbering rowboat, but a craft that once spotted on the horizon, could be landing on the beach in only 45 minutes, makes all the difference in the world.
The Gokstad ship type is quite different from the Drakkar, being beamier, deeper-drafted and higher sided. It is a workhorse to carry lots of stuff for long distances. It sails well, and rows well too, though not as well as the Drakkar, but it is emminently seaworthy. These were large craft too, up to at least 70', and perhaps larger. As far as warfare is concerned, this is the type most likely to be used as a transport for an invading fleet, with the Drakkars in front landing shock troops to seize control of the beaches and landing areas. Because of its seakindly ways, the Gokstad is considered one of the primary ships used in the Vikings explorations and long-distance voyages.
The third Viking ship type is the Knarr, which is a fairly small, very beamy and actually rather tubby craft up to about 50' long, mostly sailed, rather than rowed (small crew). This was the primary type used in trading, exploration, transportation of people, cattle, sheep, horses, and all sorts of cargo. The Cog, or Kaag is essentially a 'modernized' and more specialized descendant of the Knarr.
Now on to the Normans! While it is certainly true the Normans were not Vikings, in terms of politics, religion and allegiances, the fact is, they had only become 'French' 60 years before William the Conqueror, and ship technology in the North had NOT progressed significantly from the time of the 'classic' Vikings. It should be remembered that half the reason Saxon King Harold was defeated by William at Hastings in 1066 was because he and his army had just forced-marched all the way down from Yorkshire, after defeating (yet another!) invading Viking horde led by the Viking warlords, Harold Bluetooth, and his great pal, Thorfinn Skullsplitter. So you can hardly say the Vikings had been 'cashiered' in any way by early Norman times, or their 'technology' either! Further, their ships were more than capable of carrying horses and other gear just about anywhere they chose. Remember, horses were brought to Iceland, and the Iceland breed indicates Mediterranean blood,and these horses could not have been brought in small numbers either, or selective breeding would not have been possible. You might want to have a look at a book titled 'The Medieval Warhorse' by R.H.C. Davis to get an idea of horse transport in Medieval times (check out page 62 for a view of a Byzantine horse transport ship of 1278). Finally, it should ALSO be remembered that the English Channel is narrow, and there is no indication that William the Conqueror came over to england in just one wave. It would seem reasonable to assume that cargo and livestock ships would have made a number of trips back and forth to France to bring all that the Norman knights needed to prosecute their war against the Saxon King Harold.
Finally, the Normans did not simply settle in France and then take England. On the contrary, they continued in the path of their Viking ancestors, by invading and seizing kingdoms in Ireland, Wales, Italy, Sicily, Spain, and efforts in North Africa as well. By expanding into the Mediterranean, the Normans would have had contact with other Mediteranean peoples and their ship designs, and certainly would have taken the best elements for further expansion in the Med, which would have coincided with the onset of the Crusades.....