The show "The Bastard Executioner" paints 14th century combat as being mostly swords and some axes, with swift strikes and complex combos. Was this really what 14th century combat was like?
As for the " mostly swords and some axes", no. As u/AlviseFalier wrote in https://www.reddit.com/r/AskHistorians/comments/6pkbfc/in_medieval_battles_where_everyone_fought_with/dkr32lg/ and I wrote in https://www.reddit.com/r/AskHistorians/comments/ae3b17/if_spears_were_generally_better_than_swords_why/ the dominant hand-to-hand weapons were spears and other polearms.
Further, missile weapons - bows, crossbows, slings, and javelins - were plentiful, and often the major killer/wounder in battle. These kinds of weapons complicate battle choreography in movies and TV because they keep the opposing characters further apart, can depersonalise battle, and make it harder to safely film scenes. Thus, they're often omitted.
As for "swift strikes and complex combos", the small part of the fighting in "The Bastard Executioner" doesn't strike me as particularly swift or complex. Perhaps compared to the slow and ponderous sword-swinging seen in other movies/TV (e.g., Game of Thrones was a big offender in this regard), it's fast and complex. Fast is important, since an opponent will either (a) get out of the way of slow ponderous attacks or (b) get you first. A first attack will often not hit, since the opponent can avoid or block or parry it. Follow-up attacks, or feint-attack combinations, or parry-riposte combinations are more likely to hit. But these are not very complex. Fancier stuff (e.g., Obi-wan spins) are often figments of the choreographers imagination rather than anything resembling real fighting.
For a look at early 14th century fighting technique, as described by Royal Armouries Ms. I.33, an instruction manual for how to fight with sword and buckler, see
For technique like this at speed, used in competitive fighting, see
Fighting on the battlefield might not be quite this complex, since movement can be less free (with the need to maintain formation, and other people getting in the way).
Were battles always disorganised gore fests?
There would be a certain amount of chaos and disorganisation, and often gore. Medieval literary descriptions of battle, such as can be read in The Travels of Marco Polo, suggest gore and an element of chaos:
So then, as they were thus singing and playing, though ordered and ready for battle, the great Naccara of the Great Khan began to sound. And that of Nayan also began to sound. And thenceforward the din of battle began to be heard loudly from this side and from that. And they rushed to work so doughtily with their bows and their maces, with their lances and swords, and with the arblasts [crossbows] of the footmen, that it was a wondrous sight to see. Now might you behold such flights of arrows from this side and from that, that the whole heaven was canopied with them and they fell like rain. Now might you see on this side and on that full many a cavalier and man-at-arms fall slain, insomuch that the whole field seemed covered with them. From this side and from that such cries arose from the crowds of the wounded and dying that had God thundered, you would not have heard Him! For fierce and furious was the battle, and quarter there was none given.
Historical descriptions could be more gory, as in this :
When William, on whom the Saracens turned their attack, saw this, he knew that his life was forfeit. Manfully he bore up against all assailants, and cutting in pieces many, sent their souls to Hell. Yet though his horse had succumbed and his own feet were cut off still he continued to lop off the hands, heads, and feet, of such as attacked him.
And then, after sustaining many blows and wounds, with blood gushing out, and overwhelmed by the stones of his assailants, he, a most glorious martyr, breathed out his soul, that sped forth to assume its crown.
from the 13th century Chronica Majora by Matthew of Paris.
Note that both describe the use of missile weapons (stones in the latter).
As for disorganised mobs, we see this in neither art nor literature. Formations matter, and discipline and maintaining formations matter. For examples from art, and modern examples (from riots) of ordered-but-chaotic fighting, see my reply in https://www.reddit.com/r/AskHistorians/comments/ebya8k/did_a_soldiers_swordsmanshipskill_really_matter/
Would a skilled swordsman be able to penetrate chain mail easily?
No. Mail was made to keep weapons out, and it more or less did its job. Mail backed by, or sandwiched by, textile armours could keep out arrows from high draw weight bows, and could stop lances. Of course, not all mail is the same - some is more protective, at the cost of greater weight (thicker rings and/or smaller diameter rings). Some mail would not stop arrows and lances. But it would not be easy to penetrate by swords - swords were one of the things on the battlefield least threatening to mail, and if mail could not stop a sword from penetrating easily, it might as well not be worn (saving money and weight).
There is a shortage of good quality tests of exactly what it takes to penetrate mail, but what good tests there are (e.g., the tests reported in Alan Williams, The Knight and the Blast Furnace, Brill, 2003) suggest that lances, polearms such as halberds, and high-powered bows and crossbows were bigger threats to mail and wearers of mail than swords.