Eliminate superfluous words; you are saying things with too many words. To demonstrate such a transformation, I will take your first line through stages:
I strongly believe that such type of dieting as vegetarianism should be pursued by everybody,
I strongly believe diets like vegetarianism should be pursued by everybody,
I strongly believe everybody should pursue diets like vegetarianism,
What is pursuing a diet like vegetarianism? It is just being a vegetarian!
I strongly believe everybody should be a vegetarian,
Eight words versus fifteen words, and it conveys the same sentiment. Take one more out with a minor reprhasing and a better word (embrace, or adopt).
I strongly believe everybody should embrace vegetarianism,
And there you go. less than half your original words.
Identify what is important: "I strongly believe", "everybody", "should become (or something indicating a transformation) "vegetarian(ism)".
"That" is almost always cuttable.
Simple phrases are better than verbose: "such type of dieting as vegetarianism" is just "diets like vegetarianism". The only diets like vegetarianism are more strict than vegetarianism (like Veganism), so if you are trying to make a point, just say your least restrictive diet that accomplishes your goal: Instead of "diets like vegetarianism" say "vegetarian diets" or "Vegan diets".
Later you say "Last but not least," you could say "Finally,".
You say "a correctly balanced vegetarian diet", the word "correctly" adds nothing at all, if it isn't "correctly" balanced, then it isn't balanced! If you say "a balanced diet", readers will assume there is a correct way to balance a diet, and also that not all vegetarian diets are balanced.
What does "has long been considered [to be] of high nutritional value" mean? It means "is known to be nutritious". Doesn't "for both children and adults" mean "for children and adults"?
You have a lot of words that are adding no meaning. Presumably you are doing this for emphasis, but it does the opposite of what you want it to do; it draws out the sentence so it reduces the impact. Get rid of them, rephrase if necessary, and you can probably cut this down to 50 or 60 words, leaving you more room to make another point in your argument. Or to then add adjectives or emphasis where you most want it.