Initially, it appeared that TurboTax Online Premier could handle investments. It successfully retrieved my 1099 forms from my brokerage. However, when I moved on past the income and deductions steps, and entered a review step, it said there was missing sale price information for one of my trades. I basically day traded last year so I have hundreds of entries. Some of them triggered wash sales. In the past, those were problems with TurboTax and had to be entered manually.
I looked like it tried to pull up all of the trades into a 1040 worksheet or schedule or form (maybe 8949?) for each trade. The web page couldn't handle all this; tried with both Chrome and Firefox. Both reported a web page taking a long time and prompted me to wait or close it. After waiting many times, Chrome one time was able to pull up the form. But when I tried to enter the data, the form behind and the entry forms didn't line up. It was completely unusable. Also, it was inside a much smaller form on the web form. Really bad design.
Does the desktop version handle hundreds of investment trades any better?
Yes, but up to a limit. Please see this TurboTax FAQ for limitations on stock transactions and a work around for TurboTax: How do I enter a large number of stock transactions in TurboTax Online?