No it doesn’t. Cutting your hair short can make your hair appear thicker because the hair shaft is often thicker toward the base. Furthermore short hair cuts tend to look thicker than longer hair styles because the hair spikes upwards without a parting. So if you want your hair to appear thicker you’re better off having it cut short.
There are other advantages to short hair with regard to preventing hair loss
Short hair is less likely to get caught and pulled – if you tie your hair back tightly it can train the hair to grow closer to the scalp surface until it eventually falls out.
It’s often easier to apply hair growth lotions to your scalp if you have short hair. This is especially true if you have shaven hair.