You have to use your stabilizing muscles and core more when doing single leg exercises. Over time, it will become much easier and with higher weight. You will develop an improved neuromuscular connection. Everyone is different, so I assume your trainer feels this is a good starting point for you. However, I'd inquire about plans of eventually going to barbell squats - they can be intimidating for some at first or semi-contraindicated if previous back issues, etc. Goblet squats are great, in my opinion.
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