Is it normal for my morning sickness to persist in the 2nd trimester?

No. Women's symptoms vary. It might not be abnormal for some, but it is not typical for most women to have morning sickness beyond 13-16 weeks. So women having persistent or severe nausea or vomiting should inform their OB providers. A blood test for electrolytes (eg, sodium, potassium) and other studies might be ordered, referral to a nutritionist might be advised and treatments might be suggested.
This can happen. Although morning sickness may stop by the 13th or 14th week of pregnancy (or sooner), there are some women who continue to experience it during the second trimester. Ginger can be very helpful to quell. It. Take care.