Disclaimer: I neither watch, nor particularly like, Game of Thrones. Beyond the first two books, and watching this scene via the internet, I have little engagement with the series.

First things first, I want to address the problem inherent to answering these kinds of questions. Medieval combat is not something that can be divorced from the context it was waged in. The types of battles fought in medieval Europe varied significantly by time period and geographical area. Even campaigns within relatively similar contexts, such as the early and late Hundred Years War, can be substantially different from each other. This is important because Game of Thrones is set in a fantasy world with its own lands, society, and traditions. While much of Game of Thrones is vaguely War of the Roses-ish (it lacks things like gunpowder weaponry) it periodically draws aspects from different centuries within the Middle Ages for its inspiration. Battles cannot be viewed entirely in a vacuum, either, they must be considered in their broader context. One cannot just discuss the battles of Crecy or Poitiers without also discussing the English battle-seeking strategy of the early Hundred Years War, as that overall strategy explains much of the context and reasons behind those particular battles. Game of Thrones can handwave many of these problems away in favor of interesting characters and plot developments while real history cannot. This is particularly important to remember in this context as while most of GoT is late Medieval / Early Modern in style, this battle does not fit into that context at all.

I suppose the tl;dr of this is that a battle from Game of Thrones can never be historically accurate, because Game of Thrones is not history. It's accuracy must be on some level judged based on how it fits into the world constructed by the books, not whether it would have happened the same if it happened in our own world.

As a battle from the Hundred Years War or later, this would be a complete mess. Part of the reason for this is pretty obvious up front, though, in that nobody is wearing proper armour. The strategies of late medieval warfare are rooted in the equipment they had, and the protection offered by full plate as well as the restrictions (while often overstated in popular culture, that stuff was still heavy, especially if you planned to wear it for hours on end). As a battle this feels like it fits best in the wars of Edward I, the bit about shooting your own troops while they're in the melee is even straight out of Braveheart. I'll try and address how it fits into this context, but you're also welcome to disagree and feel it is instead attempting to copy a different era of warfare.

With regard to shooting into a melee...while there's some evidence it may have happened it certainly wasn't the plan and not really what archers were for. Archers were in general used to fire opening salvos against each other. Ideally you would use it to thin or weaken your opponents ranks in preparation for a charge, but they could also just end up in an archery duel. Until the Hundred Years War archers generally made up only a small portion of most medieval armies, even Edward I the 'father of the Longbow' didn't use a majority archers in his armies. In cases where you would be shooting into a melee, usually you would try to have your archers shoot the back ranks of your opponents army rather than into the middle of the fighting. Armies would generally engage with deeper ranks than shown here. The biggest concern was that your soldiers would break and flee the field, so you wanted to make sure they had troops behind them pushing them forward into combat. An unsupported chaotic melee like that in the show is 100% not what you want. Anything can happen in that situation and you do not want that at all. The outcome of a medieval battle was unpredictable enough without intentionally making it more random.

The circular shield formation thing is almost farcical. I have never heard of it in the context of a medieval battle, and I'm not sure any medieval general would even dream of attempting it. For one thing, it requires insane coordination and training to perform, something that was often in short supply on a chaotic medieval battlefield. Furthermore, why would you even want to surround your opponent like that? That's the opposite of what a medieval general wants. He wants his opponent to break and run, so his cavalry can chase them down in the pursuit and hack them to death one at a time. Cornering your opponent and not giving them any ground to flee can result in disasters like Courtrai, it is best avoided if at all possible. Encircling armies happened, although all the examples I can think of are in the context of Crusader warfare with Muslim armies doing the encircling, but that kind of careful, completely surrounding your opponent in a shield wall, was not a thing. The kind of encircling you did see was more about attacking from the flanks/rear rather than completely preventing someone from fleeing. Letting them successfully get away alive was a different matter, but you wanted them to run because runners aren't fighting.

There's also the matter of Sansa's miraculous arrival in the nick of time. I'll straight up confess that the similar event from book 2 of GoT is a large reason why I no longer read these books, it annoys me too much. Generals had scouts, they generally kept track of where other armies are. Many campaigns feature interesting stories of armies sneaking out from under enemy observation and avoiding detection, but these only make sense because people were looking for those armies. If an allied force was only a matter of hours away it would generally involve a complete reconsideration of the time of attack and, in the case of the defender, whether to remain in the field at all. Sometimes forces did show up partway through a battle, the late arrival of a French contingent at Agincourt is sometimes blamed for Henry V's execution of the prisoners, but it usually wasn't completely out of the blue. In the Agincourt case, they knew the forces were coming, they were just really late and the French army felt like it had sufficient forces to win the battle / it couldn't wait longer for the arrival of a few more troops. For the most part both sides were invested, for different reasons, in not fighting a battle where another army showed up halfway through. See also not adding more random elements to your battle plan.

There are relatively few instances of battles resulting in the immediate taking of a city/fortress without any kind of siege, although they do happen. That said, if medieval armies had giants they could use to bash gates open they might have been more common.

For general reading on English warfare, especially the period of Edward I, I recommend Michael Prestwich's Armies and Warfare in the Middle Ages.

Anne Curry is great for all things Hundred Years War related.

While Maurice Keen's England in the Later Middle Ages is a tad old, it's still excellent. His edited volume Medieval Warfare also includes some great introductory chapters.

If detailed battle discussions with maps and all is your thing then Kelly DeVries Infantry Warfare in the Early Fourteenth Century is amazing.

I think this goes without saying, but this episode only aired yesterday. The question is fine, but if you are opening this thread, you should expect SPOILERS in the text. If you plan to watch the episode and have not yet, you might want to close this tab!

As a specific followup question; are there historical examples of head-on cavalry-on-cavalry charges from the Medieval Era? I understand there was one by the river at Cannae a thousand years earlier in which both sides subsequently dismounted, but that doesn't so much fit the era being imitated by GoT. If there are examples, how did they go? Were they ever as truly headlong as that one in the episode?

Alongside these same lines, in the show we saw a pile of dead bodies almost ten feet high. Is this sort of thing representative of an accurate Medieval battle?