Is it permissible to travel to America to obtain equivalence for a medical degree, which includes traveling to take a practical and oral exam and interviews, and what should I do if they do not approve a visa for my father or brother to accompany me?

Traveling to non-Muslim countries is not permissible except out of necessity or dire need, provided one is safe from temptation, such as traveling for study. For a woman to travel, the presence of a husband or a mahram (definition) is a condition. If a mahram is not available, she must seek another way to obtain qualifications in her own country.