Does anybody else this problem?
No, other people understand bicycle frame geometry and know why it's possible a well-designed frame can and will allow a person's foot to contact the front tire. These people don't have a "problem".
What do you do when you want to pedal out of a turn or during a turn?
I proceed to continue to steer the bike and pedal in whatever manner necessary so as to not fall over or hit anything.
Back pedaling IS an option.Although you say you "love the bike", you should be warned that if you have the "mindset" to crash, you may very well become unable to operate the bike confortably in tight quarters. This could become a 'real' concern if you decide to add full fenders.
As others have mentioned, normal people "deal with toe-overlap" - but this may not apply to you. Good luck.