Oxfam says that some 670,000 people have left Syria since the crisis began in March 2011 but numbers have risen recently. In Jordan alone, the number of people crossing daily trebled in the past week.
The cold weather has seen an increase in the number of those suffering respiratory infections and pneumonia, according to the aid agency, which is distributing mattresses, blankets, heaters and gas oil to help new arrivals.
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The bank was taken out of public hands in May 2017.
The 18-year-old was shot in each arm and leg during the attack in the early hours of Sunday.
Concerns have been raised that the life-saving service will not be doctor-led as most services are across the world.