You can put two independent clauses together with a comma. There is nothing wrong with that. When they talk about too many commas, they usually mean more than two clauses separated only by commas.
The above sentence could have been written as:
I was so determined to learn that I hired a tutor who started with a 90--minute lesson.
Notice that 'that' is substituted for the comma.
Also, some dependent clauses could be strung together:
"I got up out bed, brushed my teeth, took a shower, had breakfast and started of for work."
This is legitimate, as the complete thought involved short dependent clauses that have to do with what I did to get to work.
This would not be acceptable:
"I got out of bed, when I noticed the sun out, I looked out the window, out on the lawn I noticed a man standing."
Here, the phrases were longer and not as closely connected. The commas detracted from the understanding.
Hope this helps.