Portadown driver Turkington is now two points behind Gordon Shedden in the points after coming home second in race two and third in race three.
Carrickfergus driver Chris Smiley also made it into the points in race one.
Subaru's Ash Sutton took a double victory, with Honda driver Shedden winning race three.
Turkington, 35, qualified second on the grid, getting the jump on pole-sitter Jack Goff to lead away.
However, debris pitched the BMW driver into spin at the Bomb Hole mid-distance, which dropped Turkington to seventh place.
However, the Northern Irishman made good progress in race two, pulling off a clever move into Nelson on team-mate Rob Collard to clinch second place.
In the reverse grid race three, twice former champion Turkington moved up to fourth from seventh, sneaking onto the podium on the final lap when leader Andrew Jordan retired with a mechanical fault.
Smiley, starting from 22nd in race one, stayed out of trouble with an excellent drive to come home in the points in 14th place.
On the soft tyre in race two, Smiley was running within touching distance of a second points finish before tyre degradation consigned the 25-year-old to 17th place
Smiley's race three was ruined by Rob Austin, who dived up the inside of the BTC Norlin Racing driver, clipping his rear wheel and forcing him down the order, with Austin also taking out championship leader Sutton in the incident.
Turkington moves up to second in the standings as a result, two points behind Shedden with Collard two points further back in third while Sutton lies fourth, 11 points off the lead.
Daniel Harper claimed a double victory in the Ginetta Junior category which moves the Hillsborough teenager up to joint-fourth in the standings.
Belfast's Charlie Eastwood also picked up victory in the first Porsche Carrera Cup race, taking the championship lead after coming home second in race two.
Colin Turkington earned two podium finishes in the British Touring Car round at Snetterton to move into second place in the championship standings.