there are 2 ways pvt company can get exemption
1st Fulfill 4 conditions
2nd Small Company
if pvt company is covered in any one of them 1st or 2nd they will be exempt
so in your case small company will get exemption 
 

Sir then in that case every company will claim exemption posing as a small company if it satisfies the condition for turnover and paid up share capital.
Will it not render the clause for private company meaningless?

intention of the law is to give benefit to small companies also and private companies which are within limits from point of many criterias 
these 2 are seperate exemptions
its fine if companies will get exemption in any one of them
 
further caro is applied by auditor’s not companies
companies wont plan to avoid caro practically speaking 
and turnover condition is not something which can be planned easily as they cross 2 crores in preceding year they wont be small company