What is the difference between 通常 and 平時 ?Feel free to just provide example sentences.

They have a very subtle difference. Both of them mean “usually, often, normally”. My interpretation is that 通常 is used for describing a bigger context and making a subjective statement while 平時 is used for describing personal habits and hobbies.

Subjective statements:

Ex. Smokers often die from lung cancer.
食煙嘅人通常死於肺癌。O
食煙嘅人平時死於肺癌。X

Ex. Boys usually like robots.
男仔通常鍾意機械人。O
男仔平時鍾意機械人。X

When you mean “under normal circumstances”:

Ex. Normally, water evaporates into steam at 100 degree celsius (unless it is not at sea level).
通常水會喺攝氏一百度蒸發成水蒸氣。O
平時水會喺攝氏一百度蒸發成水蒸氣。X

When you want to talk about hobbies, use 平時:

Ex. I usually do sports.
我通常會做運動。X
我平時會做運動。O

Ex. I usually play the piano (as a hobby).
我通常鍾意彈琴。X
我平時鍾意彈琴。O

Nevertheless, they can be interchangeable when describing habits:

Ex. Chinese usually eat with chopsticks.
中國人通常會用筷子食飯。O
中國人平時會用筷子食飯。O

Ex. My dad usually drives to work.
我阿爸通常會揸車返工。O
我阿爸平時會揸車返工。O

Ex. I usually go to bed at 11.
我通常十一點瞓覺。O
我平時十一點瞓覺。O

Ex. I usually do sports with my friends. (habit)
我通常會同朋友一齊做運動。O
我平時會同朋友一齊做運動。O

Conclusion:
Statements - use 通常
Hobbies - use 平時
Habits - use either