From her public search for a husband to attempted suicide and hospitalisation, the Irish singer has had a turbulent year even by her own standards. But her new album is joyous, brave and her most accessible in years

Visitors to Sinéad O'Connor's website are greeted by the message "This site contains adult themes and is for age 18 and over" and a photo of the singer. She stands there naked, her body covered from chest to thighs by an acoustic guitar, nothing less than you might expect from an artist whose every action seems designed to polarise. O'Connor has spent the past 25 years forcing onlookers to take sides: depending on your tastes, she's either an attention-seeker whose capricious behaviour has taxed fans' loyalty for years, or one of pop's bravest artists.

Lately, she has been particularly brave – or capricious. She's releasing her first album in five years, How About I Be Me (And You Be You)?, but the occasion has been eclipsed, to put it mildly, by events in her personal life. Since last summer, they have run roughly thus: in August she whimsically advertised online for a boyfriend, found one and quickly married him, split up with him and was reconciled (twice), attempted suicide and was hospitalised, either for depression or bipolar disorder (professional diagnoses have differed; she herself thinks she has post-traumatic stress disorder after being abused by her mother when she was a child).

That's the past seven months in a nutshell, and it's been a bruising time, both for her and her new husband, Barry Herridge, a therapist employed by the Irish government to counsel young substance abusers. Is she ever told she's brave? She looks at me incredulously across a dimly lit dressing room backstage at The Graham Norton Show, where she will later play her new single, The Wolf is Getting Married. "No." Silly question, really. Over the years, O'Connor has said she's been consistently misunderstood by the media, and "nasty people" generally.

Recently, she has taken to her website to express her feelings, and some of her posts are wrenching to read. This, from a few months ago, is typical: "Since my first record all that has happened is I get treated like a crazy person, in a world where crazy is used as a stick with which to beat someone. It's very nasty … And loads of people then in ur life think its ok to treat u like shit and dismiss u as 'mad'. Its very sore."

Yet when she's happy, her writing is full of mischief, as in the boyfriend "ad", which she originally wrote as an article for the Irish Independent. In it, she wittily and explicitly detailed what she expected of a man (hence the "adult themes" warning). It was a moment of lighthearted openness that, she says now, the media won't let her forget. "The reaction I had from people … the usual, 'She's mental,' and that it was the culmination of my lack of self-esteem. 'Send social services round!' It made me depressed about being me."

O'Connor is governed by the need to express herself – it's clear she gets much of her strength from putting her feelings in the public domain, whatever the repercussions. At the moment, she's "in the middle of a wrangle" with an American magazine for that very reason. "It's over an interview with a cunt who wrangled an interview out of me when I was depressed recently. He said it'd be a respectful interview about mental health, but he ended up saying I was crazy for saying I liked sex."

Bespectacled and dressed in head-to-toe black – hoodie, leather trousers, T-shirt imprinted with Jesus's face – she's sipping coffee and girding herself for her encounter with Norton. The wire-rimmed glasses are a newish addition. She's 45 now and says she needs them, but also uses them as a shield when she's onstage: "They're like sunglasses – they put a bit of distance between me and the audience. I'm very shy when I sing."

Nonetheless, she's looking forward to being in front of the cameras: "Today has been a good self-esteem day." It started with being interviewed on BBC Breakfast, where she was asked unexpectedly knowledgeable questions about the album. One of the programme's staff told her afterwards that the presenters loved the album; nine hours later, she's still smiling.

It made a change from the day before, when she was doing promotion in Amsterdam. "I've had 25 years of people treating me like a crazy person, and I've not been well recently. And these journalists were trying to find evidence of my extreme madness."

It would take a leap of the imagination to find anything mad about her right now. She speaks so quickly and quietly that some of her words get lost in the Dublin-accented flow, but she's cogent and very droll. (On her four children, who all have different dads: "Father's Day is a bit of a revolving door at my house.") And she has great faith in How About I Be Me?, which has received the warmest reviews she's had in a decade. One American critic even pronounced it a "full-on revival", and, in effect, it is. It's her first "pop" album since 2000's Faith and Courage, and the most accessible thing she's done since.

In the intervening years, there have been a record of traditional Irish songs, a reggae album and a collection of religious songs, all of which sold poorly and rendered her almost a niche artist. Most of the media coverage in that time focused on her personal life – her coming out as a lesbian (and subsequent back-pedalling) and her marriages (Herridge is her fourth husband). Her voice, always a thing of silvery, spectral beauty, was almost forgotten.

Many of the 7 million people who bought her 1990 breakthrough, I Do Not Want What I Haven't Got, jumped ship years ago, but those who remain will find her voice is breathier but still true, and her ability to write singular songs undiminished. The new album's hymnlike closing track, VIP, is one of the most striking pieces she has ever recorded. Whether or not it gets into the charts, O'Connor is happy with it.

"A lot of people say, 'You destroyed your career by tearing up a picture of the pope'" – on an infamous apperance on US TV in 1992 – "but I define success differently in this spiritually bereft business. To me, it's 'Can I be myself?' I could stand in the street and sing and get enough to pay the bills. I don't need millions of dollars. That was why I tore up the pope's picture – I knew I had enough money that I didn't have to marry a man with a very small penis to get the bills paid. I don't want any man to have control over me. And that is success."

In fact, men figure prominently on the album, but strictly on her terms. The record is dedicated to her novelist brother Joseph and to her former boyfriend, and father of her youngest child, Frank Bonadio (with whom she seems to be on the most amiable terms: "He wants to meet girls, he wants me to advertise him. I want to put him on dating sites, but he's too shy"). Moreover, some of the songs are about being euphorically in love. From the opening track, 4th and Vine – a skiffle-ish bit of fun narrated by a bride preparing for her wedding ("I'm gonna put my pink dress on and do my hair up tight/I'm gonna put some eyeliner on, I'm gonna look real nice") – to the stately ballad Old Lady, about having a crush on someone who "makes me laugh like an idiot, not be so serious", the record shows O'Connor's tender side.

Her unwavering campaign to hold the Vatican accountable for child abuse in the Irish church, and her frustration that other Irish artists refuse to join her, are represented on Take Off Your Shoes and VIP respectively. Her own favourite, though, is I Had a Baby, a subdued piece of electronica that addresses "the difficulties a child suffers from parental abandonment". She allows that it's about her second-youngest son (there are three sons and a daughter, aged six to 24), but adds: "I don't want to talk about the songs, because it takes away the magic." She now has a good relationship with her children's fathers, she adds.

Nonetheless, what sticks after several listens isn't the painful songs but the joyous ones, in which she luxuriates in being close to her lover. But the assumption that they're about Barry Herridge is incorrect.

"It was written between 2007-09, and recorded in 2010-11. I was going out with [Bonadio] and writing these songs. My creative process is quite slow. I hear melodies in my head while I'm washing the dishes and I allow my subconscious to do the work." The album's title came more recently, though. "My experience for 25 years" – ever since her 1987 debut, The Lion and the Cobra – "is that I've had a lot of people's opinions of me shoved down my throat. Then I wrote [the Irish Independent] articles last summer because I wanted to join a dating agency and I wanted to save the registration fee, so I wrote my own ad."

The article, and accompanying tweets about being sex-starved, while written with much self-deprecating humour ("My sh*t-uation sexually/affectionately speaking is so dire that inanimate objects are starting to look good … I must end now, as I have a hot date with a banana"), attracted derision. She was accused of humiliating herself – and, of course, some assumed she had done it to court publicity for the new album. One night, feeling depressed, she was lying in bed and the phrase "how about I be me and you be you?" came to mind. As a summation of her current mood – embattled but defiant – it couldn't be better.

Presumably, the turbulence of recent months will inspire a new set of songs? She shakes her head. "I write from a different place now. There's nothing in the recent past I want to write about." Yet the recent past hasn't been all bad: not only did it produce her husband, there was a Golden Globe nomination for the song Lay Your Head Down, from the film Albert Nobbs. And when she sings, people still respond.

A couple of hours later, now wearing a leather corset that displays Jesus's face tattooed across her chest, she performs The Wolf is Getting Married. A technical hitch means she and her band have to run through it twice; both times, the audience's applause is loud and warm. When she leaves the stage, her face is the happiest it has been all night.

(c) 2012 The Guardian