Ms. Kaur, who is 25 and Punjabi-Canadian, is used to the ups and downs. In the three years since her blockbuster “Milk and Honey” was first self-published and later picked up by Andrews McMeel Publishing, she has dealt with all the issues other women face on Instagram and off: comparisons, aggression, bullying. But she has also built a community and an audience there in particular, with 1.6 million followers.