Is it normal for my skin to be this blotchy after hair laser removal? (photo)

What you are seeing is pigmentation from the laser.  It is a temporary phenomenon.  You should go back to the physician (hopefully) or center and ask for a consultation with the dermatologist who supervises the center.  They should be able to offer you complimentary treatment.  You will need to avoid the sun during this time. 

Thanks for your question and posting pictures.  I'm sorry this happened to you.  You should immediately follow up with the supervising physician where the treatment took place.  I am guessing that they used the wrong settings for device for your skin type and sun exposure.  You should probably find a different place.  The hyperpigmentation may fade on its own or it can be successfully treated with the correct laser and settings.  If the dark pigmentation persists for more than a month, seek out a PicoSure FOCUS treatment.  FOCUS is excellent at brown pigment reduction and is safe for all skin types.
If LHR hurts that badly, you are in the wrong place.  To be safe, you should seek out a laser vs. IPL to help avoid complications.  Almost all newer lasers are safe for all skin types and have features built in (such as sapphire cooling tips) to make them more comfortable.  Some of the newer technology is also more efficient, allowing for lower energy settings to be used with better results and less discomfort.  With modern equipment, your treatment should only be mildly uncomfortable, but not painful.  We never use numbing cream for LHR.  We want our patients to be able to tell us if they feel pain or an unusual level of discomfort.
We recommend finding an experienced and reputable provider in your area with updated equipment.  A consistent overlapping technique is the key to avoiding patchiness in LHR.   Calibration and power of the laser is important for a safe, comfortable and effective treatment.  Find out what laser they will use and if it is under a maintenance contract.  If their laser isn’t calibrated on a regular basis or is underpowered, your treatment may be painful or inefficient. After a series of 6-8 initial treatments, you may need a touch up every 3-5 years.  Best of luck.