Is a small contraction of side walls of nose during normal breathing is really normal?

Maged85014,
Thanks for the question.  What you are referring to is called internal nasal valve collapse, and there are options to address it if it needs to be addressed.  I am guessing that you had anatomic issues (Deviated septum, concha, turbinates) restricting the airflow to both nostrils, and these were preventing you from being able to collapse your nasal sidewalls.  After these anatomic issues were addressed, strong inhalation is giving you nasal valve issues.  The biggest question is whether this contraction is just something that you notice when you look in the mirror (not causing you any flow restriction), or if it is making inhalation difficult.  If it is the former, you are in the majority of people, as most strong people, given an inhalation strong enough to elicit it, will have some nasal sidewall collapse.  If it is the latter, there are surgical options to add strength to the nasal sidewall so that your breathing is improved under physiologic conditions (though with a strong enough inhalation, you will likely be able to collapse your nasal sidewalls).

You describe something called Lateral Wall Insufficiency of your nasal cartilages and soft tissues. This may represent a weakening of your alar valve cartilage or other related issues. Sometime nasal valve correction or a procedure called Repair Vestibular Stenosis is needed to address this concern. Patients who suffer with Lateral Wall Insufficiency sometime benefit from the temporary use of Breathe Right nasal strips at night for improved airflow. If you are experiencing sleep related difficulty, you may want to try this. If you get relief, you may be a candidate for corrective surgery. Hope this answer helps.