Barcelona defender Bronze felt the strain on Steph Houghton, who led the national side for eight years, became increasingly difficult to deal with.
Houghton is not in the squad for the continental tournament on home soil and Arsenal's Williamson has been named as the replacement captain for the competition.
Bronze hopes a collaborative approach will mean Williamson does not have to change as a person or player to fulfil the demands of the role.
"As a captain, everybody knows Leah really well," said Bronze. "Sometimes people put too much pressure on the captain, I've seen it with Steph previously.
"But Leah is still the player that she's always been in this England squad, and she'll remain the same for us - a good person to have around, a good person to have on the pitch.
"It will be a big moment for her but she's the same as everybody else.
"We have plenty of leaders on the pitch and in the squad who may not even step on the pitch - and it's important that we remember that as a team.
"It's not just on one person's shoulders, it's falling on 23 players."
Bronze elaborated on the pressure felt by Manchester City's Houghton, who could not suitably prove her recovery from an Achilles injury to be picked for the Euros.
"I spoke to Steph a lot in previous years," Bronze said. "She told me about being given the captaincy so young and what pressures it put on her.
"She didn't know quite who she was as a captain and as a leader. It can be quite difficult to handle that.
"The media especially are the ones that are putting the pressure on captains because from inside the squad and even the coaches, you don't change as a person, people don't see you any differently.
"You're the captain because you're the person that you are. It's not because you want to become something else or change who you are. It's for that reason only.
"Steph felt that pressure a lot. Although she probably didn't admit it at the time, I know it was difficult. I was her right-hand-woman for a long time and I could see it.
"For Leah, it's important that we have other people step up to the plate to take the pressure off her and know what it means to be a leader in this team."
Bronze played the full game as England beat Belgium 3-0 in its first warm-up game for the Euros last week, with friendlies against the Netherlands and Switzerland up next.
The Lionesses then begin their Euro 2022 campaign against Austria on July 6 rated among the favourites to go all the way.
Bronze added: "Really good [preparation so far], the excitement is definitely there.
"There are so many players in their first tournament, people like Lauren Hemp, Chloe Kelly and Ella Toone.
"Just seeing their faces when they found out they were picked and that excitement just brings the buzz back to players like myself and Jill [Scott] who have regularly gone to tournaments."