The archbishop of Canterbury has admitted to having doubts about the existence of God and disclosed that, on a recent morning jog with his dog, he questioned why the Almighty had failed to intervene to prevent an injustice.

In a light-hearted but personal interview in front of hundreds of people in Bristol cathedral last weekend, Justin Welby said: "There are moments, sure, where you think 'Is there a God? Where is God?'"Welby quickly added that, as the leader of the world's 80 million-strong Anglican community, this was "probably not what the archbishop of Canterbury should say".

Earlier, BBC Bristol's Lucy Tegg, the interviewer, reminded him of the weight his words carried. "You have a remarkably prominent role within the faith community around the world," Tegg said.

"I've noticed," Welby quipped.

Tegg then asked: "Do you ever doubt?"

Welby replied: "Yes. I do. In lots of different ways really. It's a very good question. That means I've got to think about what I'm going to say. Yes I do." He added: "I love the Psalms, if you look at Psalm 88, that's full of doubt."

Welby suggested that his doubts were a regular occurrence, by recounting the recent morning run with his dog.

"The other day I was praying as I was running and I ended up saying to God: 'Look, this is all very well but isn't it about time you did something – if you're there' – which is probably not what the archbishop of Canterbury should say." He added: "It is not about feelings, it is about the fact that God is faithful and the extraordinary thing about being a Christian is that God is faithful when we are not."

Later in the interview, Welby said he was certain, however, about the existence of Jesus, even talking about his presence beside him.

"We know about Jesus, we can't explain all the questions in the world, we can't explain about suffering, we can't explain loads of things but we know about Jesus," Welby said. "We can talk about Jesus – I always do that because most of the other questions I can't answer." Asked what he did when life got challenging, Welby said: "I keep going and call to Jesus to help me, and he picks me up."

Welby, who was appointed to his post in November 2012, has frequently made known his feelings on a wide variety of issues. He has raised questions about the "inexplicable" increase in energy prices, which he has said are putting strain on low-income households.

He has criticised government changes to welfare and targeted payday lenders, saying that he wanted to "compete Wonga out of existence", although it was later pointed out that at the time the church was an investor in Wonga.

He has also offered wholehearted support for the consecration of women bishops, describing the rejection in 2012 as a "grim day".