Does shared hosting slow my website down?

It can, and the effect is inconsistent, which is what makes it hard to diagnose:

• You share the machine - CPU, memory and disk are divided between many sites, so a busy neighbour affects you.
• The symptom is variance - the same page fast one minute and slow the next, with no change at your end.
• Server response time is the tell - if the server takes a long time before sending anything, the problem is upstream of your page.

Before paying to upgrade, confirm the delay is really the server and not oversized images or blocking scripts.