I am a little confused. How did the board state in your image rapidly snowball into 40 damage on turn 4?
They have 1 card in hand. They need to trade off at least two or more minions into your Rat to get past it (assuming they didn't have a God-draw minion removal as their last card.
Your turn on turn 3 would have removed another minion with your hero power.
Alternatively, you play Wildfire and ping. And then you play the minion that deals your HP to all enemies (I believe it is 4-cost so should work out fine here).