부탁하다 means "ask," "beg," or "request," and 잘 means "well," so 잘 부탁합니다 literally means, "I beg well," which sounds as if the person is bragging about his begging abilities. Instead, when you want to ask someone to do their best for you, the "traditional" Korean way is to say one of the following:
- 청을 잘 들어주십시오. - Please grant my request.
- 잘 돌봐 주십시오. - Please do your best for me.
- 많이 도와 주십시오. - Please do everything you can for me.
잘 부탁합니다 supposedly comes from a Japanese expression, but it does not translate well into Korean.