Did you miss where the player walked the dog, returned, and it has not come back around to that players turn? The poster was saying that it took so long for the other players to take their turns that the dog-walker knew he had time to walk his dog.
That's a DM problem.
Like, I play the same game, and there is no way in hell I'd have a player go off and walk his dog mid round, or the other players take so long that this would be seen as in any way acceptable.
In my games, I give my players roughly 6 seconds (I'll start a slow count back from 6 after a few seconds) to tell me what they're doing after announcing it's their turn, or their PC takes the Dodge action and their turn ends.
You spend the time it's not your turn (while everyone else is having their turns) watching the board, and planning what to do when your initiative comes up.
I'll show some leniency to new players (it's one of the DMs jobs to teach new players the game after all) by pausing the countdown to explain a class feature or spell. After 3rd level you're on your own though.
Turns only take forever, if the DM permits it. We're not playing Rolemaster here (with percentile rolls, parry bonuses, charts indicating roles on other charts, in different books, upkeeping stun rounds, bleeding, penalties from wounds etc). On most turns you're casting a spell (and there is a save involved) or making an attack (an attack roll is involved) and then rolling damage. Each turn shouldn't take more than 30 seconds to resolve, a minute at absolute most (Action surging fighters rolling several attacks etc).
DM: OK (checks initiative order) Steve (Fighter 6), you're in combat with 5 orcs. It's now your turn.
Steve: I'll attack the Orcs, toggling GWM 'on'. Makes attack roll (hits, rolls damage, drops Orc 1). Makes attack roll (misses). I'll now attack with my bonus action from GWM. Makes attack roll (hits, rolls damage, drops another Orc). Now I'll action surge, taking the attack action. Makes attack roll (misses). Makes another attack roll (hits, rolls damage, drops another Orc).
DM: Nice work, you kill three Orcs! (checks initiative order). OK... Bob, you're up next.
Bob: (Looks up from phone) huh? What's going on...
DM: (Frowning at Bob but otherwise silent) Six... five... four...
Bob: (looking down at table, sees 2 Orcs in melee with Steve)
hurriedly - I'll cast Scorching Ray, using a 2nd level slot. Makes 3 attack rolls, rolls damage for each hit.
Etc.