South Yorkshire Police inspector Stephen Ellis told how a crush started outside the stadium on 15 April 1989.
He told the jury he feared someone would be killed before a gate was opened relieving the throng outside.
Ninety-six fans died following the crush at the FA Cup semi-final game.
Insp Ellis, who was in charge of officers meeting Liverpool fans as they arrived in Sheffield, told the inquests that at 14:30 that day it was "just a normal football match".
He told the jury he had escorted a group of around 400 Liverpool supporters from Wadsley Bridge train station into Leppings Lane in time for the 15:00 kick off.
But by 14:40, he noticed things had changed and it was "mayhem", the inquests at Warrington, Cheshire heard.
He said: "They were running towards me, away from the turnstiles, pushing their way through, shouting, screaming "get the match delayed" that sort of comment."
A group of officers on horseback tried to block access to the turnstiles, the jury heard, which was effective for a short while.
At 14:46 a police Land Rover arrived in Leppings Lane, he said.
Mr Ellis said he climbed on to the roof of the vehicle and shouted through a loudspeaker at fans to stop pushing.
"People [were] getting crushed against the turnstiles and down Leppings Lane as far as I could see, which was maybe 400m before the road bent out of sight.
Profiles of all those who died
'It was wall to wall supporters coming towards me, coming towards people in the turnstile area that were getting crushed.
He said the horses were having difficulty as they had no space either.
Mr Ellis told the inquests he saw a man "trying to protect his daughter".
"He had got his feet up against the the wall four feet off the ground. He was pushing back with his shoulders and pushing back with his legs.
"He stood up with his arms by his side, unable to move, pinned up against the wall near the turnstiles."
He also saw a young fan dive under a police horse, he told the jury.
"He cleared something like three metres without touching the ground and landed on his hands on the other side and then climbed up and started pushing to try and get into the turnstiles."
The jury heard he was worried somebody was going to get killed or seriously injured and that even if more officers or horses had been deployed they would not have helped the situation.
"It was too late; there were too many people there to have controlled by that time and still they were coming down Leppings Lane and Catchbar Lane."
He told the inquests at sometime after 15:00 the crowd cleared after Gate C was opened.
He told the jury: "I didn't even know this gate existed, let alone had been opened.
"I didn't know where they had gone."
Mr Ellis said he continued to watch for a "couple of minutes" and fans were still "diving over the turnstiles".
"There was no reason for them to behave like this; they were not getting crushed.
'It were [sic] like they were just diving over, climbing over - any way they could as if they had all been let in for free."
The jury heard that officers in Leppings Lane then closed the perimeter gates leading into the turnstiles.
Insp Ellis told the jury fans then started "coming to us vehemently angry".
"They demanded to be let in to the ground.
"They [couldn't] understand what authority a police officer has, or I have, to deny them entry when they have got a ticket."
He said he was confronted by fans outside the Leppings Lane turnstiles who claimed they had been sold forged tickets from a "boy" before the match, he told the jury.
He said it appeared they had been photocopied and coloured in with a highlighter pen or crayon.
The inquests continue.
Liverpool supporters at Hillsborough were "gripped by mania" and "frantic" to get into the ground as kick off approached on the day of the disaster, the jury at the new inquests has heard.