Is it true than men can perform mutah while being married to one to 4 women? And why such rule is allowed?

Yes that’s true. If the wife, when conducting the marriage contract, did not put a conditional on him that he must get her permission to do Mut’ah, he is allowed to do Mut’ah even if he is married. Now this doesn’t mean he should abuse this law. He must be responsible and not do something that ruins his marriage.

Some reasons why this is allowed:

1- A man’s sexual needs are different than a woman’s needs. Sometimes a man becomes desperately in need to relieve himself but his wife is not around or she is ill. He may be traveling or she is traveling (relatively long trip). Or she may he sick or pregnant and unable to meet his needs. Islam has put a way for him to relieve himself in the halal way.

2- Some women may have different needs and they cannot get married permanently (due to various reasons).