My Bike doesn't seem to fit itself
The geometry of this frame looks odd. 622mm wheels are too large to fit comfortably in a 51cm frame. Usually sizes this small are made to compensate with very steep seat tube angle to let the rear wheel forward and very shallow steering angle to avoid toes clipping front wheel. This bike has been made the other way around and that is why it looks and feels strange.


With saddle as low as shown in picture, the frame is too high for you. With horizontal top tube, there should usually be about hand's width of seatpost showing. The saddle should be roughly horizontal.
Longer handlebar stem or less swept back handlebars will help.
This is a problem with all small frames with large wheels. Is the fork the original one? Some older bikes came with steep steering angles and very high fork offsets. The offset eliminates some of the problem with toe overlap but makes steering twitchy at high speeds.
32mm tires are the absolute maximum size to fit inside short reach brakes. Normally the kind of brake is used with 23mm to 25mm tires.


Edit: @Will's comment led me to the idea that this bike could have been a 650B or at least converted into one. That would reduce the standover height and allow wider tires and also explain the strange-looking angles. Smaller wheels would lower the bottom bracket, so saddle would still look too low.