The RAW rules simply don't have a way to handle a 70-creature battle without it being excessively cumbersome. However, there are three things that I think will help you on this.
Unearthed Arcana: Mass Combat
This Unearthed Arcana provides rules for tracking mass combat. You organize groups of creatures together and treat them as single units, with some very simplified rules for what happen when two groups fight. The cool part about this rule set is that the player characters are not a part of the unit groups - they can act outside of the group restrictions for the rest of the NPCs and swing the tide of battle by eliminating units from an enemy group, casting spells, or otherwise contributing to the battle.
If your characters have interest in a more war-game style of play and want to be involved in the whole battle, I'd go this route and let them each take command of a group of militia. These rules are super fun, but don't make the fight much more complicated than a normal fight of a party versus several enemies. The only downside is that the party has to learn a new set of (very simple) rules - this will be easy for some parties and a challenge for others.
Rather than trying to keep track of the entire battle as a whole, treat the majority of the battle as a backdrop. Have in mind a rough idea of how things are going to go each round without player intervention (for example, maybe 1d6-1 guards and 1d4-1 bandits die every round), and each round narrate how the battle is going around them. However, focus specifically on the slice of the fight immediately around the party. For example:
As you and the militia charge in, fights quickly break out around the perimeter of the camp. You hear the shouts of battle and the ring of sword against sword, and see both bandits and militia taking wounds, retreating, and falling in battle.
You charged in with the right flank of the militia attack. Ahead of you, you see five bandits here, here, here, here, and here. Additionally there are two bandits here and here armed with crossbows. With your are three militia here and here, as well as the militia captain here. Roll initiative.
Until they finish their fight, you don't really need to keep track of anyone outside the scope of that battle at all. You could periodically give a brief descriptive update ("looks like the battle on the south flank is going poorly for the militia, but a group of militia to the east have successfully eliminated the sentries and are fighting their way into the camp."), but I'd stay away from anything more than that. As enemies and allies die, introduce new NPCs on either side as seems appropriate ("three more bandits stream in with swords drawn").
For a great example of this, see the Battle of the Hornburg in the second Lord of the Rings movie. There's definitely a sense of a huge battle, but we don't spend an inordinate amount of time tracking the battle as a whole. Instead, the focus of the narrative (the camera) zooms in on the section where our heroes are acting. We'll see Aragorn, Gimli, and Legolas repel orcs trying to climb their section of the wall and save their elf allies (in a small and manageable fight), and then the narrative zooms back out for an update on the status of the battle as a whole. After their fight on the wall, they realize that the battle is going poorly at the gate and rush over to help defend the gate, at which point they go into another smaller and more easily-managed fight. Aragorn and Gimli only ever face maybe 4-10 orcs at a time, with more coming in to replace those that fall until the fight is over.
DMG: "Adjucating Area of Affect" and "Handling Mobs" rules
On page 249-250 of the DMG, we see some additional rules for mass numbers of mobs that I'd recommend you read. Whatever tactic you take, this should help speed up the pace of battle. "Adjucating Area of Affect" provides some easy guidance for determining how many enemies are hit by an Area of Affect Spell. "Handling Mobs" provides a way to calculate the success of a large number of creatures making an attack without having to roll dice for each. If nine bandits shoot at your Barbarian and they need to roll at least a 13 to hit, then you can just say that three of them hit without having to roll dice.