I would definitely upgrade to a larger fan. 110 is close to the recommended minimum for your space and doesn't account for teenager marathon showers or lack of a window.
Residual moisture and humidity in one thing but this streaks are being formed WHILE the shower is in use. I have a similar problem in my 4x8 little bath even though I have a 110 cfm fan. Part of my problem is floor to ceiling glass tile on all four walls that condensates.
Panasonic has a 190 cfm fan. I like theirs because the fans are quiet. My neighbor bought an 80 cfm that was .3 sones. Almost silent. I went larger for the same 4x8 space with a 110 cfm at .8 sones because I was considering a steam shower(we did a joint stacked condo bath project so I ordered both fans). Some of the big box fans are cheap but sound like busted chainsaws at 2.5 sones when running. My neighbor was all over the quiet factor. I wouldn't have cared about sones if it wasn't for her.
Panasonic even has a 290 cfm fan but that might suck one of the kids up in the vent.
Also, you may or may not need make-up air in the house if you go large enough. Not sure what the cut off is for make up air but you don't want to suck furnace and water heater chimney gas back in the house. For some reason I remember 500 cfm but don't quote me.
You may not be able to change it now but how the fan is ducted can be part of the problem. Flexible not as good as rigid. More elbows worse than fewer elbows.
What you can change:
Increase fan cfm
Allow more air in. Cut the door up an inch. Also, you might consider through the wall vents. Rather than place ugly vent covers on a door, consider one or more pass through holes in the drywall between studs. 12x12? Or 9x9? You can cover both sides with a flat vent cover. I've done this for friends who got dinged by a home inspector for make up combustion air in a utility closet. I placed the vent up high and turned the the grates so you can't see through in to the wall cavity from ground level. Drawback: lack of privacy - may not be a problem upstairs in a hallway but not so good off a living room or kitchen.
Add a timer. I have mine on a 60 minute digital count down timer from a big box. 20-30 bucks. Honestly, a 120 minute timer might be better. Or if you want to get fancy, a switch with auto moisture sensing.
Tell everyone to leave the door open when finished. I have this argument with my wife. She is worried the dog will get in to the trash can so she closes the door. She leaves at 5am and when I use the bathroom there is still a bit of condensation on the window and walls at 7am. And that's with a 110 cfm fan on 60 minutes. I tell her to place the trash can on the toilet if she is worried. Note to self: buy a trash can with a lid.
If I remember correctly, those Panasonic fans are rated for continual run use. Like 100,00 hours. I think They can be used as home ventilation.