Ras al-Jedir on the Tunisian-Libyan border is the closest route out for those leaving western Libya.
Many of those fleeing are Tunisians, carrying with them stories of escalating violence.
Some Libyans have also entered Tunisia, planning to seek asylum.
The Red Crescent said on Wednesday that more than 5,700 people had fled across the border over the past two days.
Nazanine Moshiri reports from the Tunisian side.
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