"Each drop lasted a couple of seconds. There was a loud bang. That was when everybody got really scared.
"The worst part was during the second drop. It felt like a rollercoaster. The drops were that long. Everyone was screaming and crying".
However, another passenger said there were four drops in altitude, the Daily Mail reported.
Gregory Giagnocavo, who was on board travelling to Heathrow, told BBC Radio 4's PM there was no warning.
"This was capital T Turbulence," he said.
"There were four dramatic drops in altitude. The plane just fell what felt like 1000 feet. A split second it did that again, again, again, four times.It was terrifying."
Giagnocavo earlier tweeted a photo of a stretcher at the airport after landing.
"Severe turbulence mid-flight, injured passengers. Emergency landing Ireland. Pure chaos," he said.
Js_anghera tweeted: "My United Airlines flight from Houston to Heathrow has just plunged 4000 feet above the Atlantic Ocean".
The victims are said to have suffered lacerations, bruising and minor head injuries. A member of the crew is believed to have a fractured wrist.
Nottingham student Leslie Chi told friends on Facebook that she "really thought I might die" as the plane dropped.
Conor Brady told the Daily Mail most people were asleep when the plane when the turbulence struck.
He said: "All I remember is feeling my seat fall from beneath me and my back hitting the cabin roof, as I fell I managed to use my hands to block the blow of the chair in front.
"A lot of people were asleep and they probably had no idea what was going on until they were falling back down, afterwards a few people screamed but everyone around me quickly got back into their seats and put their belts on, holding onto their arm rests really tight.
"My friend was in sheer panic and I was in complete shock, all the time this was happening it still felt like the plane was falling and shaking."
Some of the injured passengers may not have had their seat belts on and were tossed around as the turbulence shook the plane.
The Irish Aviation Authority (IAA) said it would be investigating the incident.
Some 223 passengers and 12 crew were on board and they were met by ambulance crews and paramedics.
United Airlines said in a statement: "United Airlines is providing care and support to customers and crew of flight UA-880 which experienced severe and unexpected turbulence during a flight from Houston to London Heathrow today.
"The aircraft diverted to Shannon Airport in Ireland where it was met by medical personnel.
"We wish these passengers and crew a quick recovery from their injuries'.
The 207 passengers on the flight would continue their journey to London on an alternative aircraft, United Airlines said.
