Is my H bridge circuit correct?
I don't think so. You have all transistors marked as the same type, and you are driving each side of the H with the same control signal. 

Looks to me like you'd have a short straight from 12V to ground either through Q1+Q3 or Q4+Q2. 

Flipping the switch would then kill whichever pair didn't die when you connected the battery. 

Conceptually, you have one big problem.  You are driving the transistor pairs incorrectly.  You should be driving Q1+Q2 with the same control signal and Q3+Q4 together.

Practically, you have another problem.  From what I've read here on EE stackexchange, you will have problems switching the N-Type FETs on the high side.  You need a higher voltage on the gate than on the source to make the FET switch.

You need to do more reading on FETs and H-Bridges, then try again.

There are lots of questions about H-Bridges on this site, so you might want to start by doing a search and seeing what comes up.