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While you wait for others to share their experience on dizziness, I thought I would share this article with you that might be helpful.
-- 10 Surprising Facts About Dizziness and Vertigo:
https://www.everydayhealth.com/news/10-surprising-facts-about-dizziness-vertigo/
You mentioned your blood pressure is normal. Orthostatic hypotension can also cause dizziness and sounds similar to your symptoms. Here's some information from Mayo Clinic:

"Orthostatic hypotension — also called postural hypotension — is a form of low blood pressure that happens when standing after sitting or lying down. Orthostatic hypotension can cause dizziness or lightheadedness and possibly fainting. Orthostatic hypotension can be mild."
-- Orthostatic hypotension (postural hypotension):
https://www.mayoclinic.org/diseases-conditions/orthostatic-hypotension/symptoms-causes/syc-20352548
Have you discussed the dizziness with your doctor?

This is my personal opinion. The cause of dizziness when starting to walk seems to be a drop in BP due to different reasons. However, the extent of the problem can be reduced by larger spacing of the feet while walking that inevitably takes place in fast walk. Balancing exercises are also helpful.