The leg extension isolates the quads like nothing else does. Here is what comes closest, what each alternative leaves out, and what equipment it takes.
If you only pick one
Take the reverse Nordic curl if you train at home, and the sissy squat if you have a post to hold. Both put the quads through the lengthened position the leg extension is known for, which none of the squat variations do. The squats on this list are good exercises; they are just not a leg extension.
What it trains
Mainly: The quadriceps: the four muscles on the front of the thigh that straighten the knee.
Along with it: Almost nothing else. That is the point of the movement: the hips stay still, so the quads work on their own without the glutes and hamstrings sharing the load.
One of those four heads, the rectus femoris, crosses the hip as well as the knee. It is the head that gets the least work from squatting patterns, which is why the leg extension is hard to replace one-for-one.
Why people look for an alternative
Your gym does not have the machine, or it is taken.
You train at home and have no machine at all.
Your knees are sore under load and you want the same muscle worked differently, which is a question for a physiotherapist, not for this page.
You want quad work that does not need a machine slot at a busy hour.
The alternatives
Sissy squat
Covers
Loads the quads through a deep knee bend with the hips extended, which is the closest anyone gets to the leg extension without a machine.
Leaves out
Hard on the knees for most people, and the load is your own body, which you cannot set.
Kit
Bodyweight; a rack or post to hold.
Reverse Nordic curl
Covers
Works the quads, including the rectus femoris, in a lengthened position while you kneel upright and lean back.
Leaves out
No way to add small increments of load, and it is demanding from the first rep.
Kit
Bodyweight; something soft under the knees.
Split squat
Covers
Heavy quad work per leg, with the front knee travelling forward.
Leaves out
The glutes and hamstrings of the front leg share the work, so it is not isolation.
Kit
Bodyweight, dumbbells or a barbell.
Step-up
Covers
Quad-dominant single-leg work, easy to load with dumbbells.
Leaves out
Balance becomes the limit before the quads do, for many people.
Kit
A box or bench; dumbbells optional.
Goblet squat
Covers
Quads plus glutes in a pattern most people can load safely.
Leaves out
Spreads the work across the whole leg, the opposite of what a leg extension does.
Kit
One dumbbell or kettlebell.
Spanish squat
Covers
Keeps the shins vertical against a band, pushing the work into the quads with less travel at the hip.
Leaves out
Needs a band and an anchor, and loading it heavier is awkward.
Kit
A heavy band and a fixed anchor point.
What this page does not do
This page stops at the anatomy. It says which of the four quad heads each option reaches and what it leaves out, and it says what kit you need. It does not tell you how deep to sit in a sissy squat, how far to lean in a reverse Nordic, or whether your knees are up to either. Technique is learned with eyes on you, and a knee that complains under load is a question for a physiotherapist, not for a list of exercises.
In the app every exercise has a swap button that works the same way: it says which muscles the alternative covers, which it leaves out, and what kit it takes. Filter by what you have and the list is only what you can do.