As an HR, how do I keep both the employer and the employees happy and satisfied at the same time?

Well usually employees are not happy from HR because they think that HR is the one who is decision making authority for salary increment etc secondly the main concerns of employees are their issues directly related to HR (i.e. Salary imbursement,Pay slips,Bonuses) you can increase both employee & employer satisfaction by performing your job responsibilities smoothly & timely

Make policy and SOPs clear, Clear JD with short term goals with effective succession planning is also plays important role in increasing employee satisfaction. 

Employees are the key to every organization and should be treated accordingly in terms of Rewards, Employee Engagement, Skills & Training. If they are happy they will make their employers happy with their results.

By giving them benefits which they deserve at their levels.

You have a clear job profile so need to follow the same by analyzing all the aspects for the betterment of both.

by respecting the employment contract between the 2 parties (employer & employee) and giving everyone his right as per law 

HR Manager is the bridge between the employer and employee therefore to keep both sides happy you help them reach each other correctly.

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make a clear channel of communication and update of employee progress and company goals.

clear structure & policy of Pay scale , Bonus , increment ...etc.

As a Terminal Manager in this situation, to keep both side of the table happy you need to apply the company policy, this is why certain policy's are recorded in the SOP of employee contract when they sign

As a HR, a person can be within company policy & make employees satisfied.

Incase the employee refuse to understand, the reasons behind the policy to be explained hence, you are being fair at you place.