I ride fix gear and therefor have to pedal through every turn. I've learned a few things.

1) The bike setup. Long cranks make things worse. So do wide and deep pedals. Bottom bracket height. (Much of this is dictated by the frame but wheels and tires also play in as you mentioned above). I ride my fix gears with long (175mm) cranks. (My knees insist.) I also use the old Shimano D-A/600/105 semi-platform pedals that are pretty narrow and have the outboard bottom corner rounded so ground clearance on turns is very good. Q-factor, the distance the pedals are off the center-plane of the bike also is a factor. Cranks that bow out (for chainstay, front derailleur and chain clearance) also add in.

The typical mountain bike has a high bottom bracket but also a bottom bracket spindle and cranks that offer a lot of clearance. Pedals are usually quite wide and often deep. 175 and 180mm cranks are common.

2) Pedaling style. There are several options. Keep the bike in line with your body lean. Pedals strike at the angle determined by 1) above. Lean the bike in more than your lean. This promotes a more aggressive turn and is favored by many but pedals strike sooner. Lean the bike away. It is more of a "fight" to get the bike to turn but a sharper, more aggressive turn can be taken without pedal strike. Or rock the bike so the bike is more upright when the inside pedal comes town and leans more as it approaches the top. (This is easy and natural when out of the saddle but take a little thought when seated and is hard to do at high RPMs.)

So there is a lot you can do to avoid pedal strike on turns (besides the obvious - coasting). The bike. Costs money and may introduce compromises but there is probably a lot you can do. (Starting with a mountain bike. If you were starting with a velodrome worthy track bike there would be very little room for improvement.) And riding style. I haven't seen your riding so I cannot guess the room for improvement but the technique of leaning the bike away makes a big difference, (And every so often, someone will tell you that you are corning all wrong!)

Lastly - hitting pedals. Look at whether this is really an issue. At low leans there is little danger of crashing. I had two Peugeots set up as fix gears. Both started life with 27" wheels but I use 700c and skinny tires. Both had low bottom brackets by design. On the first I used a pedal that eventually started to fall apart after too many impacts but the demise was slow and manageable. On the second I used cheap, decently made Wellgo clipless that had plenty of aluminum in the contact area. Pedal strike wasn't a big deal. Now the bike I used to race had a very high bottom bracket and narrow cranks. Race pedals. Strike happened at big leans. Never crashed but it was heart stopping.

Look at your bike. Measure the height of the bottom bracket spindle off the ground. (My bikes - those Peugeots were under 10 1/2". My city fix gear is 10 5/8", now the lowest I will go. By custom fix gear is 10 3/4'. That old race bike - nearly 11". Can you go narrower with the bottom bracket spindle and cranks? (Look at both clearances as you spin the pedals in your highest gear and you chain alignment. It's a juggling act.) Pedals. For road riding, consider replacing your pedals with clipless and appropriate shoes.

And pedaling style/judging corners well.

Ben (who's pedaled a few miles of corners)