Cut the bottle base and squeeze the bottle
The air pressure has a major role to play in this situation. If you keep the bottle vertical, there wouldn't be any room for the air to move in as the water falls through which is the reason why you see turbulence and interruptions.
There are various ways to tackle the issue. The best method would be to punch a hole at the base of the bottom and then pour in but that is just ridiculous.
The other way round would be to share the mouth of the bottle.
Arguments against the rotating the bottle idea
Of course, as Sanya said you could rotate the bottle to create a tube which passes through the water but I see complications.
If the water is filled to the brim then you are going to have trouble getting a tube of air open through the middle (it's just difficult if you have a .
It is simply not practical to rotate a bottle. If you were to try to rotate while pouring it upside down, you can't do it that fast and there is a high chance that you are going get the mouths of the two bottles out of alignment and spill some of the water.
Thirdly, it is impossible since you cannot make a whirlpool against the force of gravity. Even if you did holding it vertically and then suddenly turning it over, you won't have it for any longer than few seconds.
Arguments against the straw idea
Keeping a straw coming out of the mouth of the bottle isn't a practical one too (Shuvam's answer).
You can't keep perfect mouth to mouth contact between the bottles because the straw is going to come in the way. If you try to get too close, the straw is going to bend and serve no purpose.
The straw is going to fall down (gravity + viscosity of water; water is flowing down, it will try to drag the straw along with it) so you would need one hand to hold onto the straw and another one on the bottle.
This is way too hard to initiate it. Let's say you place the straw when the bottle is vertical. Now, how are you going to turn it upside down? You won't be able to do it without spilling some of the water.
My impractical solution
In case the OP was accepting such impractical solutions, then my solution as said earlier would be to puncture a hole into the base of the bottle so that air can flow from the top for the bottle which is filled with water. On the bottle which is below, I would make two holes in opposite directions near the top so that air can escape from the bottle which is being filled without disrupting the flow.
That can be upgraded too. Cut the base and add a piston (could be any household utensil - maybe a glass stand?) and push the water down. There is no need for air to enter since the piston is pushing the water down.
If needed add some waterproof tape around the mouths of the two bottles to keep them in place? Just kidding.
My practical solution
My practical solution would be to pour the water into the bottle by keeping it at an angle. Initially being almost horizontal and slowly increase the inclination. In this solution, there is always a gap in the mouth of 'both' the bottles for the influx and outflux of air (influx for the bottle pouring the water, outflux for the bottle receiving the water. Moreover, you can transfer almost all of the water without spilling any. Another advantage of this idea is that you can correct the inclination as the water keeps flowing to give the maximum efficiency. You can see the rate at which the water is being filled with your own eyes and hence can alter the angle.