The Red Cross in London is running evacuation centres for people who had to flee the 24-storey apartment building razed by fire.
Twelve people are confirmed dead and many people are unaccounted for.
Red Cross spokeswoman Penny Sessions told Rachel Smalley people are very concerned about friends and relatives.
"But we did trace a child who'd lost its parents and parent who'd lose a child, managed to put them back together, and in other cases it's just people waiting for information," she said.
"It will take time, it is a very distressing time."
Sessions said all day people have been bringing in food, water, clothing and supplies for babies.
"The other things that have been going is seeing just how many people have been trying to help their friends and neighbours and coming out in support of each other.
"Also we've had so many doctors who came down today, medics came down offering their help."