Let the number of visits in the second month be denoted as \( x \).
According to the problem:
- The website received 9,347 visits in the first month.
- The third month witnessed three times the number of visits in the second month, which means it had \( 3x \).
- The total number of visits over the three months is given as 13,455.
We can set up the following equation based on this information:
\[ 9347 + x + 3x = 13455 \]
Combining like terms results in:
\[ 9347 + 4x = 13455 \]
To isolate \( 4x \), subtract 9347 from both sides:
\[ 4x = 13455 - 9347 \]
Calculating the right side:
\[ 4x = 4108 \]
Now, divide both sides by 4 to solve for \( x \):
\[ x = \frac{4108}{4} = 1027 \]
Thus, the number of visits during the second month is:
\[ \boxed{1027} \]