Hundreds of migrants, many of them children, have been rescued from the Mediterranean by a British naval ship in an operation that has seen more than 4,000 people brought safely to shore in the past two days.

HMS Bulwark has been operating in waters just north of Libya, intercepting the dangerously overcrowded boats in which thousands are risking their lives to flee war and poverty in Africa. The Italian coastguard said the 19,000-tonne assault ship has taken part in some of the 22 operations carried out over 48 hours on Friday and Saturday.

Not all were successful in saving lives, however. A spokesman said 17 bodies had been found on three boats during one rescue attempt.

Bulwark operates five landing craft which intercept the unstable open wooden boats attempting the treacherous journey to Italy before bringing passengers on board or transferring them to Italian craft to be taken to safety on land. The ship has joined vessels from Italy, Germany and Ireland in the international mission codenamed Operation Weald.

Prime minister David Cameron sent the ship to join search and rescue efforts after 800 people died in a boat that sailed from Libya, crammed with people fleeing Islamic State fighters who are terrorising the population and fuelling instability, earlier this month. It is estimated that 1,600 people have drowned so far this year trying to make the crossing – including children travelling without their parents.

The Ministry of Defence said the rescued migrants were clearly exhausted from their perilous journeys, which had taken many months for some, but all clearly relieved to reach safety.

Naval leaders warned that fine weather means they expect many more migrants to try to make the crossing in the coming weeks.

Commander Gavin Edward, coordinating the ship’s arrival on the jetty at Taranto, southern Italy, said: “The speed with which the Italian Red Cross, police and government officials have received these survivors has been really impressive and as a result we should be able to set sail later this afternoon.”

Inside the towering grey sides of the amphibious warship, the 450 members of the ship’s company were preparing to return to its search and rescue mission.

Bulwark’s commanding officer, Captain Nick Cooke-Priest, said the work necessary for a quick turnaround – “restocking water, medicines, humanitarian supplies and food, as well as clearing the mountain of rubbish” – required all hands on deck.

He added: “This week has already been the busiest of the year for migrant traffic, with fine weather forecast over the coming days, the surge of overcrowded boats leaving the north African coast is likely to continue. Yesterday, tragically, lives were lost, so the sooner we can re-set Bulwark and get back to sea to continue our mission, the better our chances of saving life.”

He described some of those his crew had rescued over the two days as “horrifyingly young”.

“My sailors and marines have demonstrated great skill, sense and adaptability,” he added.

Defence secretary Michael Fallon welcomed news of the operation, but called for a better solution from UK and overseas governments.

He said: “HMS Bulwark and her crew have once again saved hundreds of lives in the Mediterranean migrant crisis, offering medical assistance, food, water and dry clothes to those in need. A wider political solution is required to this crisis, but that does not detract from today’s rescue at sea.”