The phrase "dog paddle" refers to the swimming stroke or style that dogs often use. It involves moving the arms and legs in a simple and basic way, similar to how a dog moves its limbs while swimming to stay afloat. Full definition
Keeping your hand on his chest will support his legs back into the water for a successful dog paddle. (okcpetsmagazine.com)
There are some dogs that mostly can not swim; their bodies are too dense or their legs too short to effectively dog paddle. (3milliondogs.com)
She swam on, now sidestroking, now backstroking, even dog paddling for a moment as she positioned her eyes at water level and peered along the surface like some periscope. (slaw.ca)