Białkowski made his Polish league debut on 30 October 2004 for Górnik Zabrze in a 3–1 away defeat to Dyskobolia Grodzisk. He came off the bench to replace first-choice keeper Piotr Lech who had been sent off for a second yellow card just two minutes prior to Białkowski entering the field in the 69th-minute of the game.
On 9 January 2006, Białkowski signed for Southampton. His debut game was against Crystal Palace on 25 January 2006, which resulted in a 0–0 draw. He also appeared for the first time in the FA Cup, in the 1–0 win against Leicester City on 28 January 2006.
Białkowski was injured for the end of the 2005–06 season and for the early part of the 2006–07 season, having fallen awkwardly trying to collect the ball in a 5th round FA Cup tie against Newcastle on 18 February 2006. George Burley could not substitute him as the manager had used three substitutes already. Dexter Blackstock replaced Białkowski and prevented Albert Luque from scoring his first Newcastle goal.
After waiting for his chance after returning to fitness, Białkowski returned to Southampton's starting line-up against Colchester United on 16 March 2007 as a result of a three-match suspension for Kelvin Davis. He retained his place after Davis became available again by turning in some strong performances, including saving a Michael Kightly penalty in a 6-0 win at Wolverhampton Wanderers on 31 March 2007.
Białkowski was replaced by Davis at the beginning of the 2007–08 season when Southampton lost 4–1 to Crystal Palace in the Championship and 2–1 to Peterborough United in the League Cup.
On 17 March 2009 Białkowski joined Ipswich on loan until the end of the season. In his first appearance for Ipswich reserves, Białkowski was red-carded for handling the ball outside the area. He didn't make an appearance for the Tractor Boys and returned to Southampton at the end of the season.
On 28 September 2009 Białkowski joined Championship strugglers Barnsley on a week-long emergency loan. He played in two Championship matches, then returned to Southampton in League One. On 24 November he made his first league appearance for Southampton in over two years, replacing the injured Kelvin Davis in the 61st minute of the match against Hartlepool. He started the following six matches until Davis recovered from his injury. With Białkowski's contract set to expire in June, Southampton offered him an extension in May 2010.
In June 2010, Białkowski was reported to have signed for Portuguese side S.C. Braga but on 25 June he denied this, saying "I will be back at Southampton for the start of the season and I am hoping to stay with them." On 5 August, Białkowski signed a two-year contract extension with Southampton, with the option to extend it another two years.
Białkowski made his only league appearance of the 2011–12 season in a 2–2 draw with Blackpool. He was released at the end of the season after the club decided to release eleven players. Before his release, Białkowski was Southampton's longest serving player in the current squad.
On 15 June 2012 Białkowski signed for Notts County on a three-year contract, where he was named as Notts County's first-choice keeper. He made his Notts County début on 11 August 2012 in a League Cup tie at home to Bradford City. He kept his first clean sheet for the club in a 2–0 win over Hartlepool. Białkowski was praised by Notts County manager Keith Curle after he made an impressive start to the season, Curle described his performance against Sheffield United as "exceptional" and he believed Białkowski to be "one of the best goalkeepers in the division".
Białkowski kept his fifth clean sheet of the season in a 4–0 victory over Carlisle. His 10th clean sheet of the season came against Oldham, as Notts beat them 1–0. In his next game against Leyton Orient Białkowski was involved in a collision with Charlie MacDonald, he was carried off the field and taken to hospital where he was treated for concussion and a deep cut to the forehead. Białkowski made his return to action in a 1–1 draw with Walsall. Białkowski was praised by the Notts County management team in their 0–0 draw with Swindon as he made a number of vital saves. “I spoke with Pilks after the game and he said that it was the best game I’ve had for Notts County so I’m really pleased. I felt unbeatable, especially in the second half where I made a few saves and I just kept saying 'try and beat me', because I was so confident.” In his first season for the club Bartosz played 40 league games and kept 14 clean sheets.
On 15 July 2014, Białkowski signed a two-year deal with Ipswich Town. He marked his Town league debut with a clean sheet in a 2–0 win at Blackpool on 1 November 2014. He kept his place for the majority of the season as Town reached the League Play-Offs, but was left out of the team for compassionate reasons following the death of his father in August 2015. Fellow goalkeeper Dean Gerken's form kept him out of the team until he was injured in early 2016 and Białkowski returned with a string of impressive performances which saw him win consecutive player of the month awards and then the Supporters' Player of the Year for 2015–16.
During a game against Spain U-21 Białkowski was red carded after he handled the ball outside the penalty area to avoid to get the ball stolen by a striker who was near him.As of match played 7 May 2017
Football League Trophy winners: 2010
Football League One runners-up: 2010–11
Football League Championship runners-up: 2011–12
Ipswich Town F.C. Player of the Year 2015–16
Ipswich Town F.C. Player of the Year 2016–17