How can I beat Lynels?
Solution 1:
Lynels are extremely strong enemies compared to how early you'll find them in the game (for a lot of playthroughs).
If you want to know how to defeat the Lynel because you're looking for a challenge or you need to kill it I'd recommend the following things:
Have a shield equipped
You don't need to use a one handed weapon but having a shield can be extremely useful since it blocks all the damage in a risky situation. Lynels can hit extremely hard and blocking those 6-10 hearts of damage is an absolute lifesaver (especially early).
Use buff-food
The loading screen tips are absolutely right: Don't just run into a monster over and over again but try to prepare for the fight with some elixirs and buff-food. Depending on your health you can run defensive boosts but usually you should go for shock resistance, damage up and speed up.
Have lots of weapons
2000 HP is no joke. If you want to kill this thing with your melee weapons some of them will break. If you're still in the early game multiple, maybe even all your weapons will break. Keep that in mind when fighting this guy. You don't want to find yourself in a situation where your only option is using bombs.
Learn to side-hop, or backflip his attacks
ian_itor has already pointed this out in his answer but I want to mention it again since it's probably the most important thing when fighting Lynels. Dodging and counterattacking are extremely efficient. The attacks are slow and predictable. If you get the timing right you can land a lot of free hits with Flurry and it's great practice for dodging in general.
Aim for the head/use special arrows and Chuchus
If you want to use ranged make sure to aim for the head for a stagger and increased damage but also don't be afraid to use special arrows and elemental Chuchus. Freeze, shock and fire are your allies in this fight. Each element has it's own advantages and they can make the fight a lot easier when used correctly. Also be sure to not stand in a puddle and pop a lightning resistance elixir when going for a ranged method.
To kill the Lynel only two of the above mentioned points are required: A lot of weapons and good dodging. The others I've mentioned are additional help and are available from the start. I've excluded points that are unlocked since it's very possible to find a Lynel before you have access to certain advanced gadgets or powers. Obviously the fight gets easier the more you progress into the game because of better weapons, armor and powers but if you follow these steps you'll be able to kill the Lynel with little health and a bit of patience and practice (doable with Zora weapons and about 4 hearts).
One final note:
If you simply want to complete a quest with a Lynel near the objective, I'd highly reccomend checking if you can't just sneak around it since the rewards are usually not worth the trouble they cause (at least in the early game).
Solution 2:
You can stop a charge by using upgraded Stasis on them. It'll hold them for around 1-2 seconds, which should be just enough to fire an arrow into their face. When they come out of stasis, they'll be staggered and their charge will end.
My favourite strategy is to try and stun lock them. First, you need to hit it in the face with a powerful enough bow. Doing so will force it down on it's knee for a short while. While in this state, you can approach them and press 'A' to mount them. This allows you to get about 5 hits in depending on your stamina and how fast you can mash Y.
Solution 3:
The strategy I have discovered allows me to run through several of the most powerful Silver Lynels in a row, without the need of a bunch of weapons.
Preparation
Complete Ancient Armour set, upgraded at least twice each. The set bonus, Ancient Proficiency, doubles the damage of Ancient weapons. If the armour is upgraded four times, it also has the highest defence in the game.
A high damage Ancient weapon, of any type. I had a 60 damage Ancient Battle Axe++.
A fair number of arrows, of any type. A good damage bow doesn't hurt, and you'll get a new one off the Lynel.
Some food or an elixir which increases attack or defence isn't a bad idea. Environmental protection is another option, if you'd rather not take constant damage if in a cold or hot area.
Combat
Run away from attacks, and keep a medium distance. Far enough that he tries to charge at you, but also not so far that he pulls out his bow. Make sure you get away from his ground stab which makes a sphere of fire around him.
Shoot arrows at his face whenever you get the chance. This will stun him. The type of arrow doesn't matter, and while bomb arrows will deal more damage at this stage, they do not make it any easier to stun.
Whenever he's stunned, run up and mount him, then attack. Weapons do not lose durability while mounted on a Lynel. Your double damage Ancient weapon will take out a hunk of his health.
Run away. It's helpful to move the camera facing forward before being kicked off the mount, to more easily keep him in view afterwards. Rinse and repeat.