Is it normal my family keeps encouraging me to drink

Don't give in to them. You don't need alcohol. They need to respect your wishes and back off.

Your family should respect your choice not to drink, they should not try to pressure you into doing things that make you uncomfortable.

Lemme guess, Irish Catholic?

Ya don't do something you don't want to. Usually when someone wants to push you into something it's to make them feel better about what they themselves are doing.

Are your family members alcoholics? Yanno misery loves company.

Check out Al Anon and Adult Children of Alcohols and Dysfunctional Families:

http://al-anon.org/?gclid=CKGVqvK16tQCFce3wAodZpQKGg

http://www.adultchildren.org/

They probably encourage you to drink with them so you won't feel left out. While there is a possibility of something deeper than that, most family relationships have this sense of ignorance. Try explaining that you don't want to and include your reasons. You might also try encouraging them to drink around you so that you can show them that you are fine with it

They sound like alcoholics who don't want to have a sober person around because it shows them up. I've known many addicts in recovery who cop lots of shit from other addicts who try and persuade them back into the addiction: in fact, I've copped it myself from nicotine addicts after quitting. It makes some people angry to see others who aren't as addicted as they are: ignore them or alternatively try & talk calmly to them about it when they're not drunk.

Maybe they want you to drink so they can get information out of you, maybe loosen you up a bit?