I read 19 chapters of Boku dake ga Inai Machi and I was watching the anime with my girlfriend, so I assumed the manga I was thinking of was that but it wasn't.

Basically I remember there's this boy who's also a fugitive or maybe he was just running away from something/someone, but got housed by a girl with glasses I believe? There may have been some time travel stuff going on here as well since I vaguely remember there was a point where he might've went back/forward in time and then back to the present again, I'm not exactly sure.

The more I think on it, what happened could've just been in the middle of the manga because I'm pretty sure it wasn't like that right away. Either way, any ideas would be appreciated!

Do you remember the demographic? Was it Shounen, Shoujo, Seinen or Josei?
I'll just throw in a couple possibilities, just in case. Unfortunetly, I don't remember if any of these had time travel themes 🙁

Tsumi to Batsu - A Falsified Romance
Ibara no Ou
Kimi no Knife
Mayonaka no X Giten
Psyren

Demographic was either shounen or seinen, almost certain seinen. I was thinking it might've been Mahou Shoujo of the End but I'd need to reread it to see if it actually is or not rip