Expressing “to conduct", "to carry out” with 进行 and 实行
The verb 实行
We use the verb 实行 
to express that something new was implemented. For example: a political program, a law, an idea.
to express that something new was implemented. For example: a political program, a law, an idea.Examples
法国对奶农实行补贴贷款制度
France has a supplemented loan system for dairy farmers
fàguó duì nǎi nóng shíxíng bǔtiē dàikuǎn zhìdù
他们在机场实行了安全检查
They have implemented security checks at the airport
tāmen zài jīchǎng shíxíng le ānquán jiǎnchá
政府将自由实行其经济政策
The government will be free to implement its economic policies
zhèngfǔ jiāng zìyóu shíxíng qí jīngjì zhèngcè
We usually use 实行 in the following collocations:
实行政策
实行纲领
实行方针
实行计划
实行规则
实行纲领
实行方针
实行计划
实行规则 The verb 进行
We use the verb 进行 to emphasize the process of carrying something out.
to emphasize the process of carrying something out.Examples
建议进行化疗
Undergoing chemotherapy is recommended
jiànyì jìnxíng huàliáo
意大利即将进行选举
Italy will hold elections
yìdàlì jíjiāng jìnxíng xuǎnjǔ
我们在进行慈善募捐
We are conducting charity fundraising
wǒmen zài jìnxíng císhàn mùjuān