from 1 to 16 there are 16 numbers so I guess you want to put every number (we have to use all numbers since there are 16 numbers and 16 squares) once but if it is that way then the sum : 1+2+3...+16 has to be 32*4 in order to do what you ask which is not the case : 1+2+...+16=4*34. So your rows have to 34 each. If it is that way. So

10 11 6 7

5 8 12 9

3 4 13 14

2 16 15 1

should be ok with 34 in each row.

In your case, using every number between 1 to 16 once to have 32 in each row in impossible.