Six British soldiers have been killed since an additional 3,300 British troops were deployed to Helmand Province in April.
Clashes in two other southern provinces over the past day have left six suspected Taliban militants dead.
Seven coalition troops and an Afghan soldier were also wounded, according to U.S.-led forces.
The coalition said heavy fighting broke out on July 7 in Zabul Province, where a suspected Taliban militant was killed and five coalition soldiers were wounded.
In the neighboring province of Kandahar, five suspected Taliban militants were killed in multiple clashes. Two coalition forces and an Afghan soldier were also wounded in one clash, in the Panjwayi district, some 35 kilometers southwest of the city of Kandahar.
In Kandahar itself, police on July 7 raided several Islamic religious schools, known as madrasahs, to check whether any of the students were Taliban militants.
Some 25 students who were not carrying documents are still being held.