The instinctive drowning response is a climbing response. What you see are twin splashes as the person's arms flail wide looking for something to grab on to. If they find it the automatic response is to grab it, climb up, and stand on top. The person doing this is panicked and has massively more strength than you think. If you are swimming and a young child does that to you, you will go under water. If you panic, you will get the same reflex, and will drown yourself.
Therefore if you wish to rescue someone who is drowning, you have 4 basic options.
1. The most common response is to go to them and try to rescue directly. If the water is shallow enough that you can stand, letting them climb up you can be OK. Otherwise this is suicide. Frequently one drowning leads to multiple others, and this is exactly how they get started.
2. Keep your distance and hand them something to climb on. They can't see a pole to grab it, but if they touch, they will climb, which gives them air.
3. Wait until the drowning person is unconscious. Then go to them and tow them in. If you can't stand in the water, and don't have anything to do 2) with, this is recommended.
4. (For trained people only.) Swim up behind them, grab them from where they can't grab you, and lift them up to get air. Talk soothingly to them until they relax, and then tow them in. If they get hold of you, DIVE so they let go, back off, circle around, and start again.
If you do not have lifeguard training then I strongly recommend against trying #4. Unless you know how to do the eggbeater you won't know how to hold them out of water. And if they grab you, you have surprisingly little time to make the correct response before you are drowning as well.
Oh yes, and another data point. Drowning is the #1 cause of death in toddlers. It is quiet. It is fast. Beware of the backyard pool.