Independently from what your personal idea of grittyness involves - the question is how
that feeling is evoked. If we are talking about grittyness which is induced by mechanics it may be hard to look at PbtA and seeing the obvious mechanics as a factor.
If you - on the other hand like in more traditional games - have many stats and more or less an extensive book-keeping about the current state of the characters and the world you could leverage these to tighten the thumb screws on the characters (players). If you e.g. use time and resource management rules you could leverage this mechanic to make the game feeling more gritty: Think of being on the 9th level of a dungeon and your last arrows are shot, your swords are broken and your last torch goes out - just now.
Or think of the accurate bookkeeping of time which leads to calulations about the lasting effect of spells and the time slot needed to do x.
One concrete mechanic from older editions of DnD which obviously induces feelings of grittyness is level drain: Your are thrown back to a previous state with less power in a currently extreme dangerous environment.
I am only acquaint with some PbtA systems and there resource management is abstracted away - of course if your even abstract resources are gone you start thinking. The same goes for the classical HP management. But as far as I know something like level drain is nowhere established.
If you take DnD and Dungeon World there are rules for healing and recovery which could be leveraged to mechanically increase grittyness. Examples given in the DMG could be translated to Dungeon World.
But from my understanding of the PbtA philosophy the best mechanical screws for grittyness in these games are GM (especially hard) moves as a result from reaching a turning point towards the unexpected in the story (mechanically as a result of rolling a 10-) and snowballing (where one bad consequence could be only the first in a chain of bad things to happen). In a gritty environment loosing one characters complete food rations when the knapsack goes down the crevice may actually be a soft move because there are much more harsh consequences (hard moves) to be afraid of.
Another lever to pull for more grittyness is the clock mechanics from Apocalypse World which is used (with some iterations and modifications) in some PbtA games: Having these clocks in front of the players may be a mechanical way of visualizing pressure on the players which may have a comparable effect to the stats of trad games.
If you think about the roots of DnD and in consequence OSR it is obvious why mechanics played a big role in inducing grittyness in these kind of games: DnD was a game between different players - one being the GM and the others playing characters. To let the players (playing the characters) feel some pressure you could fumble around with their stats. Whereas PbtA is more about the characters and the story you have less mechanical levers to pull but build a gritty atmosphere through narration.
PbtA is as gritty as you make it.