Migrants try to stay afloat after falling off dinghy in Mediterranean on Friday
The Italian coastguard said 2,000 people were saved on Friday and a further 3,000 on Saturday.
More than 100 people drowned, including an eight-year-old boy.
Spring conditions make it easier for small boats and rafts to make it across the sea to Italy and Greece.
The Malta-based Migrant Offshore Aid Station tweeted that their crew had “never seen anything like it”.
Migrants rest on the Malta-based Migrant Offshore Aid Station
Migrant crisis: Key locations before and after
Tue, April 4, 2017
In these composite images, a comparison has been made between a scene at a key location during the height of the 2015 migrant crisis last year and the view there now
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Aid workers help migrants up the shore after making the crossing from Turkey to the Greek island of Lesbos on November 16, 2015 in Sikaminias, Greece
In one incident, the organisation’s ship the Phoenix rescued 134 African migrants after they fell from a rubber dinghy 15 miles off the coast of Libya.
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The pace of arrivals has exceeded anything we’ve seen before in the Mediterranean
The crew were preparing to transfer them to a safer “rigid hull” inflatable when one man fell into the sea and accidentally dragged in others.
The Phoenix continued searching despite being overloaded with 450 migrants.
Meanwhile, two other rescue ships, Prudence and Aquarius, operated by Medecins Sans Frontieres, rescued 1,000 on Friday.
Migrant rests after being rescued in the Mediterranean
The Libyan coastline remains a hotspot for rescues. At least 97 migrants died last Thursday, when their boat sank. Just 23 were rescued.
And in late February, 87 bodies washed ashore in Libya.
Joel Millman, spokesman for the International Organization for Migration, said in March that the “pace of arrivals has exceeded anything we’ve seen before in the Mediterranean”