Learn English – less happy or less happier

Neither is quite right. In general, you would probably want to say either

Carbohydrates in the diet need less water in comparison to proteins and fats.

or

Carbohydrates in the diet need a smaller amount of water in comparison to proteins and fats.

At a stretch, you might possibly say

Carbohydrates in the diet need a lesser amount of water in comparison to proteins and fats.

but it sounds less idiomatic than the other two options.

You've hit upon a rather tricky part of English, the usage of less vs lesser vs smaller. In general, you can compare uncountable or mass nouns (like water) using the word less:

There is less milk in my glass than in yours. The water is less than one inch deep.

You use smaller to compare countable nouns:

My glass is smaller than yours. I want the smaller cookie, please.

In your sentence, you have the countable, compound noun [an] amount of water. Because it is countable, you need to have an article in there, and you need to use the comparative term smaller:

Carbohydrates in the¹ diet need a smaller amount of water in comparison to proteins and fats.

Alternatively, you could drop the phrase [an] amount and have just the mass/uncountable noun water (which doesn't need an article), in which case you would use the term less for comparison:

Carbohydrates in the diet need less a̶m̶o̶u̶n̶t̶ ̶o̶f water in comparison to proteins and fats.

So why not lesser?

Lesser is a more specialized case. For one thing, it can only be used attributively (right before a noun), and only with count nouns. When we use it comparatively, it most often refers to quality or importance rather than quantity:

Joe is a lesser man than Pete.
Titus Andronicus is often regarded as one of Shakespeare's lesser works.

It can be used to refer to quantity, but only with a few specialized count nouns of amount or quantity:

I like raspberries and, to a lesser degree, strawberries.
There is a lesser amount of money invested in Plan A than in Plan B.
A lesser number of citizens are voting nowadays than in years past.

However, while all of these are to some extent grammatical, only the first sounds fully idiomatic to my ear; for the rest, I would prefer to replace lesser with smaller. An Ngram comparing usage suggests this trend is widespread:

¹ The word diet is countable, so it needs an article here. You could also use a possessive pronoun: Carbohydrates in your diet.

So while you could use it, smaller probably sounds better.

X is cheaper than Y. [X =10, Y=20]

X is less expensive than Y. [X=10, Y=20]

Those two sentences mean the same thing.

Conversely, Y is more expensive than X.

Cheap, one syllable, just add ER: Cheap=cheaper

Expensive, three syllables, add less or more: less expensive than, more expensive than.

In fact, X is much cheaper than Y, by 10 dollars. Much=an adverb that modifies cheaper.

much cheaper is an adverb and goes with cheaper. It is not a little cheaper, it is a lot cheaper. much cheaper=a lot cheaper.

Adding much or a lot or a little [adverbs] to a comparative is fine.

Much cheaper, but: much more expensive.

The other meaning of CHEAP=BAD quality:

Product A is less cheap than Product B= The quality of Product A is higher than the quality of Product B.

Comparative: Product A looks less cheap than Product B.