Hi happytoes. I am so glad your daughter is getting treatment. According to page 18 of "Dr. Burrascano's treatment guidelines at http://www.ilads.org/lyme_disease/B_guidelines_12_17_08.pdf Zithromax is not recommended for young children. I don't know if this is true. Hopefully mothers here can comment. If you would like the contact information for a wonderful Lyme pediatrician, you can email me. Maybe he could help and/or consult with your daughter's doctor.
From what I understand, six weeks of the proper dose antibiotic is the minimum treatment time for an infection caught within two weeks of the bite. If symptoms are still present, further treatment is warranted. A longer infection needs longer treatment. Are you sure the doctor you saw is a Lyme doctor? I don't think 20 days is long enough. This is the best time to eradicate the infection.
I think we can all understand your mistrust of doctors!