If I understand the problem correctly, it sounds like you're sitting too far forward on this bike.
Whether you're too tall for your frame or not is difficult to diagnose without at least pictures, but you might consider that this bike is simply too small for you.
(If you're having trouble getting the handlebars far enough forward, that's a sign that this could be the case.)
Are you able to get the saddle high enough
so your knees are locked with the heel of your foot on the pedal?
You might also consider simply moving your saddle back on the rails.
That might give you a little more room.
I'd also repeat
Wadelp's suggestion - toe clips will help you keep your feet correctly positioned on the pedals.
bicycles.stackexchange.com/users/8/Goodbye Stack Exchange
You could always add some toe clips to your pedals they will keep your feet positioned on the pedals better.
Clipless pedals - like Shimano SPD or Crank Bros Egg Beaters - work great too, but they do require special shoes with cleats on them.
bicycles.stackexchange.com/users/5371/Wadelp
Ideally, the ball of your feet (the part just below the toes), should be on the pedal to transfer power to the pedals (efficiently and bio-mechanically healthy).
This is also likely causing your knees to "flare" away from the frame (like a V).
You'll want to keep your knee, toes and hip on the same vertical plane.
otherwise long term could lead to knee ligament problems.
It looks like the bike is too small for you.
At 185cm, I would think a 58-62cm frame is better suited depending on your leg length. see http://bicycling.about.com/od/howtoride/a/bike_sizing.htm and http://www.ebicycles.com/bicycle-tools/frame-sizer/road-bike/size-sheet?utf8=%E2%9C%93&u=in&r=man&h=1879.6&i=863.6&b=Calculate
bicycles.stackexchange.com/users/5574/MandoMando
You may want to try this online fit computer .
It does require several measurements, but the results are generally pretty good.
Regarding changing the geometry, you can sometimes make a small bike work, but it will never be as good as a properly-sized bike.
I know someone who rides a too-small Colnago because he's so attached to it.
He has a 120mm stem (maybe longer, can't recall), and his seat is as far back as possible.
He's around 1.83m tall, and his bike is a 54 or 55 (can't recall).
He was commuting on the bike, around 200 miles (322 km)/week.
But just because it works for him doesn't mean it will work for you.
It depends on your geometry, too.
Long legs, short arms would be worse on a too-small bike than short legs, long arms on the same bike, because you need a certain amount of leg extension to pedal efficiently.
bicycles.stackexchange.com/users/3985/Zippy The Pinhead
Try an offset seat post , that will help get your seat back a little further.
Other than that, sure, a different seat tube angle would change the spatial relationship between your rear and your feet.
But that angle doesn't change too much between the various road bike frames.
Perhaps shortening the effective top-tube length (forcing you more upright) might help the feeling - you can get a shorter stem, or different handlebars.
It begs the question -
why do you care where your feet hit?
If the current scenario is comfortable, then why change it?
Unicyclists (mountain at least) ride with the pedal just in front of their heel.
Given your shoe size (58-59), I can imagine your feet are hitting the front wheel when you turn sharply - so maybe that's the reason...
On some bikes I have that problem, and I wear only 48.
More drastic changes could be switching seat types, like a banana seat (yes, you might look silly).
Or switching bicycle styles, say to a recumbent.
But, really, if the change doesn't "feel" right, have you tried it for an extended period of time?
I'm thinking a couple hours (cumulative) - to get past the initial reaction.
Maybe you've just been riding mid-foot so long it will take a while to change what feels comfortable.
bicycles.stackexchange.com/users/174/Trey Jackson
What type of pedal are you using?
You're a relatively tall guy for a 55 cm bike.
You're almost my height, I think you would be much more comfortable on a frame that's closer to 59 cm.
bicycles.stackexchange.com/users/5237/Scott Hillson
I am 1,85m tall, have shoes sized 58-59 (European) and have classic non sloped frame with seat tube 55cm. Problem is that my feet land with their middle on the pedals, not with the front part where the toes are.  I will try to move seat more to the back, but other than that, is it possible to look for different geometry on the frame, or are almost all frames the same in that respect?
Feet on the middle of the pedals
bicycles.stackexchange.com
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bicycles.stackexchange.com/questions/13298
bicycles.stackexchange.com/users/5563/Davorin Ruševljan
The problem of the foot position on the pedal appears to stem from sitting too far forward on this bike.  This may occur when the bike is too small for the rider. One option is to move the seat backwards. Toe clips will help you keep your feet correctly positioned on the pedals.  Clipless pedals - like Shimano SPD or Crank Bros Egg Beaters -  work great too, but they do require special shoes with cleats on them. It sounds like the bike is too small, so moving the saddle back with an offset seat post could help, in addition to some other changes to the geometry. Toe clips or clipless pedals could keep your feet in the right position on the pedals. A larger bike with the right fit to your body would be a better solution though.
It sounds like the bike is too small, so moving the saddle back with an offset seat post could help, in addition to some other changes to the geometry. A larger bike with the right fit to your body would be a better solution though. Toe clips or clipless pedals could keep your feet in the right position on the pedals.