I can only answer this for Ancient Greece, so bear in mind that the story may be very different for other cultures that used closely packed infantry formations (it sounds like you've been watching The Last Kingdom, which takes place over a thousand years after the period I'm talking about here).

Before the rise of organised formations, when the Greeks praised the exploits of heroes like Achilles, battle was a tentative affair. The mass of men would hang back and perhaps shoot the occasional arrow or throw a spear, while every now and then a promachos (front-fighter) would rush forward, kill his man, and try to drag the body back to his own side. Massed melee combat would only break out if one side was pressed hard or if possession of a prominent hero's body was being contested.

With the rise of organised formations at the end of the 6th century BC, battle became more uncompromising. Encouragements to fight in the front ranks turned into legal sanctions for cowardice. Pushed on by their formation, warriors were forced to fight the enemy up close. To overcome their terror in the final moments, Greek hoplites would charge into the fray.

What exactly happened at the moment of contact is a matter of debate. Some scholars argue that both sides plowed into each other and kept on pushing until one side was literally bowled over and trampled. Others, especially more recently, have argued that it probably wasn't quite as intense as that - there was certainly room enough for individual combat, and battles do seem to have gone on for a long time (though we have no idea how long is "a long time").

What we do know is that the encounter was very bloody. The average death toll of a Greek battle was 5% on the winning side vs 14% on the losing side. That 5% shows how many people died in the melee before the losing side broke; it means that roughly the entire front rank of the phalanx died in combat.

Even so, it was indeed all about holding one's ground and waiting for the cavalry, or the reserves, or for the enemy's morale to break. Head-on hoplite combat was obviously the least efficient way to win a battle; Greek generals tried their best to find a way to break the enemy's spirit, so that they wouldn't have to rely on brute force and endurance. Warriors' reluctance to get killed probably meant that parts of the line would occasionally disengage, hold back, catch their breath, and possibly try to clear the wounded. It certainly was no confusing melee of mixed lines until one side or the other was exterminated.