A driver's license

A driver's license

Something along the lines of "My Driver's Test" or "My Driving Test" might fit better. I think I would at least say "my" instead of "a" because you're talking about your experience, not some random person's.

I had a behind-the-wheel driving test in this morning, but (I) failed it.

You can take out the second "I" if you want. Also, you can say "my" instead of "a," but what you have now is fine.

When I tried to turn to left, there were pedestrians who were walking on the road.

You should either say, "When I tried to turn left" or "When I tried to turn to the left."

Also, you have too many "were"s. You could say: "When I tried to turn left, there were pedestrians walking on the road" or "When I tried to turn left, pedestrians were walking on the road."

After When they almost crossed away, I started to drive.

I think I would use "when" instead of "after" because "after" indicates the period of time following an event while "when" refers to something happening at or during the time of another event. You could say something like: "When they had almost finished crossing, I started to drive."

That was the reason (why) I failed.

You could take the "why" out if you wanted.

The examiner told me that I had to wait until they passed ''completely.''

I think punctuation marks generally go inside quotations.

I thought I waited obviously enough.

Maybe say something like: "I thought it was obvious that I waited long enough" or "I thought I obviously waited long enough."

I have to have one more exam on next Monday.

If you say "one more" here, it could indicate that you have another exam (and possibly different one). Are you taking the same test again? If so, then I think it would be better to say something like: "I have to retake the test on Monday."

I'll do more and more carefully and pass it!!

Do more what? Drive more carefully? Pay closer attention?