How were medieval castles built in swamps or marshes without draining them?

As noted, this type of castle was extremely common.

Harburg (Horeburg/near Hamburg), the first castle at Danzig are perhaps the most famous of these. They were most often built along the Northern European plains on the South shore of the North sea and the Baltic Sea. But that is only indicating that a lot of wetlands can be found in those regions.

They were popular everywhere where there was enough water influencing the surrounding terrain to form a natural obstacle for approaching enemies. A clear cut definition and distinction between swamp castle and water castle seems difficult. Today, many of these were either transformed into more comfy residences in later years and thus lost their military character (quite late construction date exception of Bourtange in Groningen), had their marshlands drained, or they did succumb – not in the least through military actions – or in their construction itself, due to the permanent water stress upon the construction.

Then of course, Prince Philip, the Duke of Edinburgh springs from a family that had its seat at Glücksburg Castle.

Castle Vischering might look most like what we expect for a medieval castle:

But to get even more to the point of Monty Python: they did not have that much money for extensive props. So they just used an existing castle:

For comparison, picture from question:

And the original:

Bodiam Castle:

and as pictured in the movie

Bodiam Castle was used in Monty Python and the Holy Grail in an establishing shot identifying it as "Swamp Castle" in the "Tale of Sir Lancelot" sequence

The difficulty in differentiating these types of castles may be illustrated in this watercolour picture by Wilfrid Ball in "Sussex Painted" (1906):

_{–– Matthew Balent: "The compendium of Weapons, Armour & Castles", Palladium Books: Taylor, 1991.}

The often found subsequent difficulties with water damaging the construction lead to some inventions:

The motte proper was a conical mound which might vary considerably in size (between 6 and 15 m). The motte could be completely man-made—an important or even enormous undertaking—but if there was a suitable hill in the area it would be adapted by scarping, that is, cutting vertically down the sides and digging away the lower slopes. In certain cases, to avoid the shifting of materials and to provide greater stability to the motte, alternating layers of stone, peat, clay, chalk, rubble, gravel, brushwood or sand were inserted between rammed or beaten down layers of earth. Finally the whole mound was revetted with a thick coat of clay to keep out water. With this technique the builders could create a dry place in a possible swampy area, or keep the castle out of reach of the river in low lands. The base of the motte was surrounded by a ditch, which may have been filled with water. Some of the material for the motte was derived from the ditch, but in some cases additional materials were required to bring the mound up to any appreciable height.

A great majority of medieval moats were dry, but de­ pending on the natural situation, some were filled with water. A wet moat, called a douve or wet ditch, formed a very efficient obstacle against the assaulting party. However, wet moats could be something of a mixed blessing; they were inconvenient in peacetime, which meant that unofficial bridges were often erected—with subsequent argument and indecision about the right moment to chop them down in an emergency. Besides, water might dangerously erode the base of the wall, and stagnant water might be a year 'round health hazard for the inhabitants of the castle.

The water for a douve could simply be collected from rain, but because this source was unreliable, the wet ditch was very often supplied with fast-flowing water coming from a river or sea by means of dikes, sluices, watergates and derivation canals. In certain cases, wet ditches took on the proportions of a lake, a marsh or even an intentional flood.

The architect and the lord or trustee would decide together the best place to build the castle. Their choice was influenced by various strategic, technical and financial considerations, and nearly always involved a site favoring defense such as a high ground, a spur, a hill, an island or a marsh.

When a portion of ditch was dug, masons built strong wall foundations (remember that castles were vertical buildings demanding stability). In good ground conditions large flat stones were tilted inwards to take the thrust of the wall above. When the ground was less stable, masons start with a framed-up timber raft; on marshy ground they had to install timber piles driven deep.

Constantly, the master-builder had to supervise all of the construction, control alignments, check material qual­ity and so on. To all these tasks were added the construction of echauguettes, gatehouse, houses, chapel, lord's residence, dungeon, and more. And the conception and construction of the castle were even more compli­cated in mountainous sites where transport was difficult and weather unpredictable. Spectacular difficulties were met in wet or marshy sites.

Certain cities owed their growth to the installation of a royal, imperial, princely or episcopal court and became national, regional or provincial capitals. Many others were rebuilt or created because of a favorable geographical situation (for example, a passage in a marsh or between mountains, a ford on a river, a junction of important cross­ roads, or a protected anchorage) because accessibility favored the installation of trading-places and markets.

Cross-section, Tour de Constance in Aigues-Mortes (France). The huge Constance tower, completed in 1248, was a 32 m high cylinder, 22 m in diameter with walls 6 m thick. Built in a swampy site, the tower rests on solid foundations composed of pillars. The tower includes two vast vaulted chambers and a crenellated top platform with a watch-tower used as a lighthouse.
_{–– Jean-Denis G G Lepage: "Castles and Fortified Cities of Medieval Europe. An Illustrated History", McFarland: Jefferson, London, 2002.}

That means for construction that builders tried to take advantage of natural conditions: drier, raised land, like in Glücksburg, where the construction rests on a natural granite base. Simultaneously or alternatively, they used compressed earth and mixed material they tried to seal from the wetness primarily with clays and rested the foundations on wooden beams, like on a grand scale found in Venice. In both examples, the permanent difficulties stemming from the wetness and continuing to this day illustrate how difficult to maintain these constructions were and are.

The case of Danzig/Malbork castle illustrates another aspect. We can only analyse a fairly short time period in which the castle existed in the 'desired' form. The necessities for upgrades over time were already quite pronounced. The present day appearance is much more solid than the first iterations:

The Knights quickly overran the lower reaches of the River Vistula, building their first castles during the 1230s, of timber and earth banks. In a land of rivers, swamp and forest, a scanty supply of good building stone meant it was used only for foundations.
_{–– Dan Snow: "Battle Castles", Haperpress: London, 2012.}

A problem from the conditions for answering this question is the assumed absence of "terraforming". Building the castle itself is a part of terraforming. In the middle of a swamp no big building can be erected without manipulating the ground for it to be able to hold the weight of the structures above it. Further it was always desirable to ensure the surroundings keep their deterrence and obstacle properties. That means castle owners would have always liked to have some more control over the water level. To prevent the swamp or marshland to become too dry, or even increase its effectiveness by initiating a flooding.

As the fate of most of these constructions who were not real water castles on a river island or a proper lake shows, these castles in swamps or marshes were most often only a temporary phenomenon. Not only for health and safety of the occupants or the buildings themselves. Ironically such castles became often small population centres. And ordinary people often had less interest in this inconvenience of wetlands next to their housings, but generally preferred usable land close to home, so that eventually most swamps and marshes were drained, or even dried up naturally.

Further examples to investigate – not for their present but for their initial appearance would be:
Wyher near Ettiswil, Gifhorn castle, the oldest still existing Burg Oebisfelde, Lembeck, Hülshoff, Dunguaire Castle, Mespelbrunn or the sometimes suspected oldest stone castle of which no traces remain from the series of Loire castles Doué la Fontaine, Château de Sully-sur-Loire.

The Teutonic Knights had come from the Holy Land to Prussia well experienced in the building of castles, but in the savage lands of the north the environment and conditions for construction were very different. Their Prussian territories were almost literally carved out of the dark forests and swamps with their meandering rivers, and the pioneer knights were handicapped by a shortage of labour, no local building skills and few deposits of workable stone. The technology of building in brick existed and had been demonstrated effectively in Denmark, but the settled conditions and reserves of manpower simply did not exist on the Prussian frontier. The result was that the first Prussian castles resembled those of the natives that were destroyed by the Order's army. They were often hastily built from timber, usually oak, because the stone laboriously gathered from fields was only sufficient for foundations. The forts were thus rough-hewn wooden blockhouses, sometimes built upon a crude and simple stone base of undressed boulders, encircled by wooden palisades and with the lay of the land determining the contour of the walls. With so many rivers and swamps around, water defences played an important part in a castle's layout. Rivers provided a barrier in summer and roads during the winter, so the castles were often located on a promontory where two rivers met. Wooden castles were of course vulnerable to attack by fire, so during the 13th century an attempt was made to bring masons in from Germany to build stone castles. However, as noted above, stone was scarce, and in Prussia no more than five castles were converted from wood to stone by 1250, so timber kept the knightly communities alive until brick succeeded stone as a cheaper and more readily available material. The average garrison of a wooden castle would be about a dozen knights, together with 100 or more soldiers, mercenaries, native militia and servants. In time, all the permanent castles of Prussia were converted from wood to brick, but it is also important to note that wooden castles continued to be built even when brick castles were well established, but these were constructed as temporary fortresses during expeditions into enemy territory. If such a foray was likely to take some time, then it was sensible to build a temporary base rather than rely on a succession of flimsy encampments.
_{–– Stephen Turnbull: "Crusader Castles of the Teutonic Knights (1). The red-brick castles of Prussia 1230-1466", Fortress, Osprey: Oxford, 2003.}