DC square wave Transformer
First, D1 and D2 in the above diagram serve no purpose. They drop 1.2 volts from the power supply. That's it. I just wanted to get that out of the way.
Transformers are tricky. They don't just give the input voltage times the turns ratio. Transformers can only pass alternating current. Balanced alternating current. If the voltage across them has a DC component, the current on the primary side will increase in the direction of that DC component until the primary resistance cancels it out and/or the transformer saturates. You never want that to happen. For every Voltsecond you apply in one direction you have to apply another voltsecond at a later point in the operating cycle to balance it out. This keeps the current from building up in one direction.
Here's a simplified version of your circuit in my favourite circuit simulator. Notice how the current just keeps on going up and up and up.
(click it, it's a link!!)

What you need is to switch the voltage to the opposite value for at least half the cycle. Here's what that looks like:

Here, we use two diodes and two switches. The diodes let the built up current run backwards, dumping energy back into the power supply. Try playing around with that circuit. You'll find that bringing the duty cycle above 50% causes the same problem as in the previous design, the current just keeps increasing. You're applying 60 volts in one direction for more than half the duty cycle and 60 volts in the other direction for the rest which makes the current build up in one direction as it did before.
Half bridge and Full bridge topologies
The most used topology in power electronics is the half bridge. You find it everywhere from DC-DC point of load converters to motor drivers to Utility scale  HVDC power conversion Put two of them together to get a full bridge.
Here's what your finished circuit would look like. The switches stand in for MOSFETs. I'll get to the practical implementation in a little bit.
In this case, the transformer is the Load the half or full bridge drives. Everything is nice and symmetric. With a half bridge configuration you only drive the transformer with half the input voltage. You also have to make sure the capacitors on the other side don't empty out. Try reducing their size or increasing the resonant frequency of the system and see what happens. (That's why the diodes are there.)
Implementing a half bridge is pretty simple. You need an independent 12-20V power supply sitting referenced to your power voltage. Add a few diodes BJTs and resistors and that's it. Here's the circuit for driving a single half bridge. The independent power supply can be produced by a charge pump. For prototyping it's easier to just use an independent lab supply for both but for the finished product a charge pump is just the circuit above. A 555 timer, two diodes and some passives. All it has to do is supply ~10mA to the upper gate driver.
That's pretty much everything. Here's what you should do:
1 Build the High side gate driver and test that it works.
2 Build the rest of a single half bridge.
3 Test that it switches correctly
4 connect it in the half bridge configuration to your transformer with a split set of electrolytic caps on the other side and try driving something.
Hopefully it works.