When I walk onacross a crosswalk at the green light, I frequently encounter rude cadrivers coming very close to me, which islike they are about to run over me.

You can say you are walking "on" a crosswalk, but I think it sounds a little nicer to say "across", since you are walking across the street to the other side.

I don't hear people say "rude cars", but rather "rude drivers". It is the driver that is being rude, not the car. When you say the driver got close to me, the person you are talking to knows you mean the car got close to you.

On small and narrow streets, I see many cars turning around the corner without flicking on the indicator, ands a result almost running over me.

You should say "turning the corner". "Turning around" in terms of driving means, for example, you were driving north and realized you were going the wrong way, so you turned around and you are now driving south.
I got rid of "and" and added "as a result" to make it sound smoother. You use "as a result" when you are describing something, and later in the sentence telling the result.

A rude car touched the hem of my overcoat on a crosswalk.

Either "a rude driver" or "a car". In this sentence, it makes more sense though to say "a car", since that was the thing that touched your coat.

Since then, I've made it a rule to stop before crossing, in spite of the green light.

Transitional words/phrases like "since then" need a comma afterwards when they begin a sentence

Then, I sawnoticed an interesting phenomenon.

saw works too, but noticed sounds better here :)

When I stopped crossing, the cars stopped moving as well.

I took back my rule and walked onacross a crosswalk atduring the green light once as before, t. Then, the cars were slowly coming very close to me, as if they were going to run over me as beforegain.

"during the green light" sounds better than "at the green light", because when you say "during", you are saying that you are walking across at the time span that the light was green.

Split the sentences into two to make it sound better, got rid of one "as before" and put "again" to reduce repetitiveness (which is when you use the same words a lot close to each other)

Again, I followed my rule and stopped crossing despite the green light when I saw any car, then they stopped to wait for me to cross over.

I think these drivers can't make a big decision by themselves, and instead always follow stronger personsople than them, not rules.

Because they are very mentally weak, but they want to show that they are strong to the appearentsupposedly weaker personsople than them.

When you say "apparently weaker people" in this sentence, it means that to your surprise these people are probably weaker. When you say "supposedly", it shows WAY more doubt. I would read into the difference of these two words, and see when to use them.

Probably, only social psychologists know thise reason why they change their behaviors.

When I walk on a crosswalk at thea green light, I frequently encounter rude cars coming very close to me, which isthat come so close they're about to run over me.

I deleted the first "to me" because it can sound a bit repetitive.

On small and narrow streets, I see many cars turning around the corner without flicking on the indicator and almost running over me.

Since then, I've made it a rule to stop before crossing in spite of the green light.

When I stopped crossing, the cars stopped moving as well.

This sentence is in present tense while the ones before and after are in past tense; I changed the verb tense to make it consistent (but the sentence you wrote is grammatically correct).

I took back my rule and walked on a crosswalk at the green light aslike before, and then the cars were slowly comingame very close to me as if theyo run over me as beforegain.

Again, I followed my rule and stopped crossing despite the green light when I saw any car,; then they stopped to wait for me to cross over.

I think these drivers can't make a big decisions by themselves and always follow stronger personsople than them, not rules.

Because tThey are very mentally weak but they want to show that they are strong toer than the appearently weaker persons than themople.

Probably, only social psychologists know this reason as to why they change their behaviors.