The single-legged fork must truly withstand heavier bending forces than conventional forks, simply due to physics and asymmetricity.
But because of its different construction, the fork is actually stiffer than most
2-legged. Pros The top is attached like a dual crown downhill fork, which is much stiffer than a single-crown.
The wheel axle is one-piece with the bottom part, which is stiffer than a quick-release axle (which is not solid at all) and on same level as a 15 thru-axle or 20 thru-axle.
This design is used also on cars where the wheel doesn't have support from the other side.
The biggest difference is inside.
While normal forks have round tubes that slide on oil film, the Lefty uses needle bearings on a square profile (same as the front leg of Boeing airliners.): Such solution works with almost same friction under heavy side-loads, whereas sliding tubes, when side-loaded (uneven compression of legs), do get friction losses (rolling resistance wins).
Having conventional fork with one leg wouldn't work, such a fork would get stuck.
Having bearing-fitted 2-legged fork is an overkill and unnecessary weight.
The heaviest Lefty is still lighter than lightest carbon 2-legged.
You can replace the tire without removing the wheel.
Cons
If you need to remove the front wheel, you need to remove the brake caliper also.
The axle works only with Cannondale hubs.
The front of bike will look unbalanced until you get used to it.
Pre-2013 models of Lefty are not sealed, having the bearings track just under a rubber sleeve.
This required almost constant maintenance.
Post-2013 models have a rubber-sealed leg with bearings further inside and with one plain bearing at the bottom which allows for service intervals as normal forks: Previous models, as some bikers on forums complained, had more linear action: they were easier to bottom than normal forks.
So now the newer ones should have more progressive compression.
(This is opinion based, but quite a lot of it online, there are no measurements.)
A Lefty is much harder to service than a normal fork.
You need a couple of special tools, and the assembly is complicated.
bicycles.stackexchange.com/users/19624/Jerryno
My experience with Lefty goes back to 2000 when I purchased my Cannondale
Super V. One thing I've liked about it
how easy it is to 'turn off'.
There's a small dial at the top that disengages the suspension - good when transitioning onto solid road riding.
As mentioned above by another post, the center of balance is altered.
The implication of which is that it is more difficult to ride 'hands free'.
Sometimes I ride hands free with it just to practice my balancing skills.
I seem to remember that the original intent was that lefty offered a longer(higher) range of motion and thus the bike was able to handle more adverse terrain.
I'm not sure if that's actually true, but it sounded good at the time.
If you do get one, you'll find it to be a conversation piece.
Since it hasn't been widely adopted, I'd imagine that the benefits are not overwhelming.
And one last thing, I've always considered the bike to be "much more of a bike than I am a man".
bicycles.stackexchange.com/users/24422/SRQ Coder
Pro: You can run oversize and plus size tyres on a lefty.
bicycles.stackexchange.com/users/26262/Mark Rose
Like everyone, the first time I saw this fork I find it obvious that it must be weaker than conventional forks... Until I realized that conventional forks are not symmetrical at all.
One side is the spring (air or metal spring) and the other side is the damper.
This means the sides always works against each other.
The fork is teared a on its sides and it do not behave exactly the same whether you turn left or right.
The forces are not applied to the same side of the plunger and it does not slide evenly on both sides.
Having damper and spring on the same axle is a good point for the lefty.
Unfortunately we cannot do blind test on bikes, but it would surely give lefty more points.
bicycles.stackexchange.com/users/25323/bokan
Con: Because there is only one side on the fork, it limits you to using disc brakes only, since there is no place to mount any type of rim brake solution.
bicycles.stackexchange.com/users/25698/K4N3
Pro: There is no "fork," so mud and other hub-bub will not get stuck in the front as much.
bicycles.stackexchange.com/users/31400/Adam
I'd add:
Pro:probably less width overall, good for extreme mountain/trail bikingCon:center of balance is shifted off center line of tire.
bicycles.stackexchange.com/users/19676/A Bee
Looks like they save some weight, but how do they have a better strength/weight ratio than a traditional two legged fork?   With the single-legged fork the fork and axle must resist huge bending all the time, producing even more of a problem than with the 2-legged fork under uneven compression.  So how is this fork constructed and what advantages and disadvantages does this bring?
Pros and cons of single-sided forks (Cannondale Lefty)?
bicycles.stackexchange.com
<mountain-bike><fork><suspension><cannondale><lefty-fork>
bicycles.stackexchange.com/questions/30765
bicycles.stackexchange.com/users/19624/Jerryno
Advantages of a Lefty: lighter, the tire can be replaced without removing the wheel, you can run oversize and plus size tires, there is no fork so mud, etc. will not get stuck in the front as much, damper and spring on the same axle. Disadvantages of a Lefty: You can only use disc brakes, it is harder to service, it feels unbalanced at first, the axle works only with Cannondale hubs, to remove the front wheel you need to remove the brake caliper.  Car wheels are also supported on only one side, so this is not as risky as it might seem. Normal bike forks have a damper and spring on different sides, so they are not symmetrical either. The needle bearings are also more resistant to friction losses under side loads than normal sliding tubes. While this system is light, easy to adjust for different road surfaces and reduces collection of mud, it is harder to service and can only be used with Cannondale hubs and disc brakes. Replacing a tyre is easier, but removing a wheel requires the brake calipers to be removed too.
Car wheels are also supported on only one side, so this is not as risky as it might seem. Normal bike forks have a damper and spring on different sides, so they are not symmetrical either. The needle bearings are also more resistant to friction losses under side loads than normal sliding tubes. While this system is light, easy to adjust for different road surfaces and reduces collection of mud, it is harder to service and can only be used with Cannondale hubs and disc brakes. Replacing a tyre is easier, but removing a wheel requires the brake calipers to be removed too.