My goal is to sell all of my stock holdings and make a complete switch to mutual funds and index funds. Would I still be liable for capital gains tax on the increase in my stocks if I sold everything I had and reinvested the proceeds all inside the same brokerage account without taking any money out?
I assumed someone would know as I am not very knowledgable about this.
Yes, it is taxable. It’s best to start a new index and stop contributing to existing accounts. Sometimes, I pause dividend reinvestment, and surprisingly, there is enough leftover to fund another investment.
What is long-term capital gains tax? It is the tax on profits from selling an asset that has been held for over a year. The tax rates vary based on your income and filing status, with potential rates of 0%, 15%, or 20%. Generally, most individuals pay no more than 15%.
What is short-term capital gains tax? This tax applies to profits from selling an asset held for one year or less. It is taxed according to your standard income tax bracket, which could be 10%, 12%, 22%, 24%, 32%, 35%, or 37%.
Yes, you will still owe capital gains tax on the growth of your stocks, even if you reinvest the proceeds into mutual funds within the same account.
Reinvesting into mutual funds does not alter the fact that you have realized a gain from selling your stocks. The tax is calculated based on the profit from the stock sale. At the end of the year, your brokerage will provide a Form 1099 detailing the gains or losses from the stock sale, which you will need to report on your tax return. 
Sure, there will be taxes on the gains.