Why was the sword so popular in medieval times?

It wasn't popular at all.

Swords were expensive, lacked range and required space and maneuverability in order to function.

They also required intense training just to get mildly competent. Training to be a mildly competent swordsman would be a death wish.

The real weapons of the medieval time were the polearms. Forge a pointy thing, stick it on the end of long stick and have a square of 100 men stand close and point their sticks at the enemy. No swordsman would ever stand a chance against a unit a pikemen in formation.

Pikemen in position ready to receive infantry or cavalry. A charge by either would have been (and often was if attempted) futile and suicidal. The real offensive weapons—missiles (arrows, catapults, etc.—stood behind this wall.

Nor would a swordsman, or even a group of them for that matter, be able to mount a defense against cavalry whereas pikemen could present the knights a wall of death.

No knight is charging that. They could find much better options for suicide. Same with a swordsman. He'd be killed by a 2–4 meter pike before he'd even be close enough to swing or stab.

Then again, the pike was far less frequently used than missile troops. The English preferred a 3:1 missile to melee mix of their army. At Agincourt it was like 5:1. The pikemen mostly served as guard duty for missile troops. They basically formed walls of death preventing the enemy from attacking the defenseless missile units. Heavy infantry wasn't deciding any battles in the middle ages, they were literally human shields.

Swords were secondary weapons. The weapon of last resort. You draw your sword if the enemy is crazy enough to attack and somehow gets past the pikes. Same with knights, they'd only draw a sword for one of two reasons. Either their main weapon was disabled or they were hacking down undefended and fleeing enemies.

You ever wonder why some swords are curved and others aren't. Well curved swords are less likely to get embedded in the bodies of people you jack at while riding past them. The curved sword was used by cavalry against routing soldiers when you'd need to cut down as many as you could before they got a hold of their wits and reorganized. There's a scene in some movie with an officer training with his sword cutting melons. I think it was from the film Glory but can only find this video.

War Gallery: British Cavalry Sword

The thing that's unstated but quite clear is the training isn't for charging packed formations, it's for hunting down fleeing enemies, chopping at them and moving on to the next. You're not too concerned with shields, helmets or counter attacks because you're running down people that lost their wits in the heat of battle, threw down their heavy items like helmets and weapons and then ran. So the primary training is how to chop and move on efficiently, why they use melons on sticks instead of people defending themselves.

Pikemen being used aggressively, as illustrated in an episode of Game of Thrones (Battle of the Bastards).

Most casualties of war were by missile volleys or cavalry running down stragglers like this. Very rarely would cavalry charge formations or infantry would go head to head. People like to live, they run at the first chance.

This is much how we associate curved swords with eastern origins. They traditionally used more light cavalry that would use ranged weaponry then mop up the enemy like in the video once they fled. The sabre is directly modified from the scimitar.

The reason why people assume swords were popular is because swords are cool, associated with the upper classes and easier to film.

First of all, having Mel Gibson running around and swinging a giant sword just looks cool. Having him stand in a tight formation holding a 3 meter pike and mostly standing around doesn't look cool at all.

Secondly, polearms just weren't practical to carry around the house or through a market. For this reason the upper class chose to carry daggers and later swords. Time period films like the Three Musketeers show them not on the front lines as musket firing soldiers but as sword wielding frat brothers.

These Musketeers wouldn't show up in battle like this but for a film it's just fine.

Most people wouldn't ever need the sword so the focus was portability. Much like how a shotgun outperforms a pistol but a shotgun won't fit comfortably in your purse. That's what a sword was, the pistol of its time.

For the same reason why pikes were impractical as civilian weapons, they're also impractical for filming. Imagine the difficulty of having dozens of untrained extras carrying around 3–4 meter pikes, knocking over lighting and boom mics.

Then take a look at that picture again. They're holding swords. Let's check other iterations.

Just what exactly is it called, Three Swordsmen or the Three Musketeers?

Yet they're not holding muskets, they're holding civilian fencing swords. Swords are just so much cooler than anything else they kept the name but still armed them differently. It's all fake news. Crossbows and pikes, later muskets and pikes and finally just guns. Last time swords played a real impact was with the late Roman Republic armies (as the legionnaires used a short sword called the gladio).

Longbowmen delivering a withering volley of arrows (as seen in Game of Thrones episode Battle of the Bastards). But they would have been well guarded behind a wall of shielded spearmen. The Battle of Agincourt (1415) during the 100 Years' War, a British victory over superior numbers, remains the classic proof of missiles' (range weapons) superiority over close combat, either on foot or on horseback. Henry V's longbowmen (up to 80% of his troops) decimated French cavalry and other formations to the shock of French combatants and observers.