A synonym for strong hand is tight rein, iron fist and firm hand if you're referring to a firm, controlling hand.
If you're referring to physical strength, then a synonym for strong hand is powerful grip or even brute force.
A fancy word for strong is robust.
Robust is often used to describe a persons or somethings physical strength or something that is durable and suggests a sturdy, strong and healthy condition.
Other fancy words for strong are might, potent and formidable.
These words convey a sense of power, strength or influence in a more sophisticated way.
The word mighty emphasizes great strength or great power and is often used to describe individuals, forces or things which are powerful.
Potent is a word that highlights the ability to exert power, influence or effectiveness and is often used to describe arguments, actions or substances.
Formidable is a word that implies that something is impressive or awe inspiring and difficult to deal with or overcome.
The word strong is an adjective which means "having power or being physically robust," and able to withstand great force or able to endure hardship and very intense.
The words mighty, muscular and powerful are some synonyms for the word strong.
Other words you can use to say strong are stout, tough, powerful, firm, vigorous, sturdy and solid.