Assuming you want any of your like conditions to be satisfied only if visible is 1 then wrap the logic you already have in parenthesis then add the visible = 1 as shown below.
PHP Code:
$response = mysql_query("SELECT * FROM `tData`
WHERE (word1 LIKE '%".$q."%'
OR conj LIKE '%".$q."%'
OR word2 LIKE '%".$q."%'
OR username LIKE '%".$q."%'
OR lineID LIKE '%".$q."%')
AND visible = 1
ORDER BY `lineID` DESC LIMIT 0 , 10");